Problem Silent Self-stabilizing Leader Election Model: Locally - - PowerPoint PPT Presentation

▶
problem
SMART_READER_LITE
LIVE PREVIEW

Problem Silent Self-stabilizing Leader Election Model: Locally - - PowerPoint PPT Presentation

Feb 19, 2024 21 likes •1.84k views

Self-Stabilizing Leader Election in Polynomial Steps 1 Karine Altisen Alain Cournier Stphane Devismes Anas Durand Franck Petit September 29, 2014 1This work has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) and

slide-1
SLIDE 1

Self-Stabilizing Leader Election in Polynomial Steps1

Karine Altisen Alain Cournier Stéphane Devismes Anaïs Durand Franck Petit September 29, 2014

1This work has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) and the AGIR project DIAMS. Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 1 / 23

slide-2
SLIDE 2

Problem

Silent Self-stabilizing Leader Election Model:

◮ Locally shared memory model ◮ Read/write atomicity ◮ Distributed unfair daemon

Network:

◮ Any connected topology ◮ Bidirectional ◮ Identified

No global knowledge on the network

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 2 / 23

slide-3
SLIDE 3

State of the Art

Model Paper Knowledge Daemon Complexity Silent D N B Memory Rounds Steps Message Passing Afek, Bremler, 1998 x Θ(log n) O(n) ?

  • Awerbuch et al, 1993

x Θ(log D log n) O(D) ?

  • Burman, Kutten, 2007

x Θ(log D log n) O(D) ?

  • Locally

Shared Memory Dolev, Herman, 1997 x Fair Θ(N log N) O(D) ? Arora, Gouda, 1994 x Weakly Fair Θ(log N) O(N) ?

  • Datta et al, 2010

Unfair unbounded O(n) ?

  • Kravchik, Kutten, 2013

Synchronous Θ(log n) O(D) ?

  • Datta et al, 2011

Unfair Θ(log n) O(n) ?

  • D: Diameter

D ≥ D: Upper bound on the diameter n: Number of nodes N ≥ n: Upper bound on the number of nodes B: Upper bound on the link-capacity

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 3 / 23

slide-4
SLIDE 4

Our Contribution

Algorithm LE

Memory requirement asymptotically optimal: Θ(log n) bits/process Stabilization time (worst case):

◮ 3n + D rounds ◮ Lower Bound:

n3 6 + 5 2n2 − 11 3 n + 2 steps,

Upper Bound:

n3 2 + 2n2 + n 2 + 1 steps

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 4 / 23

slide-5
SLIDE 5

Our Contribution

Algorithm LE

Memory requirement asymptotically optimal: Θ(log n) bits/process Stabilization time (worst case):

◮ 3n + D rounds ◮ Lower Bound:

n3 6 + 5 2n2 − 11 3 n + 2 steps,

Upper Bound:

n3 2 + 2n2 + n 2 + 1 steps

Analytical Study of Datta et al, 20112

Stabilization time not polynomial in steps:

◮ ∀α ≥ 3, ∃ networks and executions in Ω(nα+1) steps. 2Datta, Larmore, and Vemula. Self-stabilizing Leader Election in Optimal Space under an Arbitrary

  • Scheduler. 2011

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 4 / 23

slide-6
SLIDE 6

Design of the Leader Election Algorithm

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 5 / 23

slide-7
SLIDE 7

Simplified Algorithm (Non Self-stabilizing)

Join a Tree

3 variables per process p

p.idR ∈ N: ID of the root p.par ∈ Np ∪ {p}: Parent pointer p.level ∈ N: Level

1 3 5 7 6 2 4 1, 0 1, 1 1, 1 1, 1 1, 2 1, 2 1, 2

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 6 / 23

slide-8
SLIDE 8

Simplified Algorithm (Non Self-stabilizing)

Join a Tree

3 variables per process p

p.idR ∈ N: ID of the root p.par ∈ Np ∪ {p}: Parent pointer p.level ∈ N: Level

Initial Configuration

p.idR = p p.par = p p.level = 0

1 3 5 7 6 2 4 1, 0 3, 0 5, 0 7, 0 6, 0 2, 0 4, 0

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 6 / 23

slide-9
SLIDE 9

Simplified Algorithm (Non Self-stabilizing)

Join a Tree

3 variables per process p

p.idR ∈ N: ID of the root p.par ∈ Np ∪ {p}: Parent pointer p.level ∈ N: Level

Initial Configuration

p.idR = p p.par = p p.level = 0

1 3 5 7 6 2 4 1, 0 3, 0 1, 1 1, 1 3, 1 2, 0 2, 1

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 6 / 23

slide-10
SLIDE 10

Simplified Algorithm (Non Self-stabilizing)

Join a Tree

3 variables per process p

p.idR ∈ N: ID of the root p.par ∈ Np ∪ {p}: Parent pointer p.level ∈ N: Level

Initial Configuration

p.idR = p p.par = p p.level = 0

1 3 5 7 6 2 4 1, 0 1, 1 1, 1 1, 1 3, 1 1, 2 1, 2

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 6 / 23

slide-11
SLIDE 11

Simplified Algorithm (Non Self-stabilizing)

Join a Tree

3 variables per process p

p.idR ∈ N: ID of the root p.par ∈ Np ∪ {p}: Parent pointer p.level ∈ N: Level

Initial Configuration

p.idR = p p.par = p p.level = 0

1 3 5 7 6 2 4 1, 0 1, 1 1, 1 1, 1 1, 2 1, 2 1, 2

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 6 / 23

slide-12
SLIDE 12

Simplified Algorithm (Non Self-Stabilizing)

Self-stabilization = ⇒ Arbitrary initialization

2 3 4 5 1, 1 3, 0 4, 0 1, 1

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 7 / 23

slide-13
SLIDE 13

Simplified Algorithm (Non Self-Stabilizing)

Self-stabilization = ⇒ Arbitrary initialization = ⇒ Fake ids

2 3 4 5 1, 1 3, 0 4, 0 1, 1 Fake id Fake id

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 7 / 23

slide-14
SLIDE 14

Simplified Algorithm (Non Self-Stabilizing)

Self-stabilization = ⇒ Arbitrary initialization = ⇒ Fake ids

2 3 4 5 1, 1 1, 2 1, 2 1, 1

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 7 / 23

slide-15
SLIDE 15

Simplified Algorithm: Removal of Fake Ids

Reset

2 3 4 5 1, 1 1, 2 1, 2 1, 1 Inconsistency Inconsistency

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 8 / 23

slide-16
SLIDE 16

Simplified Algorithm: Removal of Fake Ids

Reset

Reset

p.idR = p p.par = p p.level = 0

2 3 4 5 1, 1 1, 2 1, 2 1, 1 Inconsistency Inconsistency

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 8 / 23

slide-17
SLIDE 17

Simplified Algorithm: Removal of Fake Ids

Reset

Reset

p.idR = p p.par = p p.level = 0

2 3 4 5 2, 0 1, 2 1, 2 5, 0

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 8 / 23

slide-18
SLIDE 18

Simplified Algorithm: Removal of Fake Ids

Reset

Reset

p.idR = p p.par = p p.level = 0

2 3 4 5 2, 0 1, 2 1, 2 5, 0 Inconsistency Inconsistency

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 8 / 23

slide-19
SLIDE 19

Simplified Algorithm: Removal of Fake Ids

Reset

Reset

p.idR = p p.par = p p.level = 0

2 3 4 5 2, 0 3, 0 4, 0 5, 0

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 8 / 23

slide-20
SLIDE 20

Simplified Algorithm: Removal of Fake Ids

Reset

3 5 2 6 4 1, 2 5, 0 2, 0 1, 4 1, 3

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 9 / 23

slide-21
SLIDE 21

Simplified Algorithm: Removal of Fake Ids

Reset

3 5 2 6 4 3, 0 5, 0 1, 5 1, 4 1, 3

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 9 / 23

slide-22
SLIDE 22

Simplified Algorithm: Removal of Fake Ids

Reset

3 5 2 6 4 3, 0 1, 6 1, 5 1, 4 4, 0

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 9 / 23

slide-23
SLIDE 23

Simplified Algorithm: Removal of Fake Ids

Reset

3 5 2 6 4 1, 7 1, 6 1, 5 6, 0 4, 0

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 9 / 23

slide-24
SLIDE 24

Simplified Algorithm: Removal of Fake Ids

Reset

3 5 2 6 4 1, 7 1, 6 2, 0 6, 0 1, 8

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 9 / 23

slide-25
SLIDE 25

Simplified Algorithm: Removal of Fake Ids

Reset

3 5 2 6 4 1, 7 5, 0 2, 0 1, 9 1, 8

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 9 / 23

slide-26
SLIDE 26

Simplified Algorithm: Removal of Fake Ids

Reset

3 5 2 6 4 1, 7 5, 0 2, 0 1, 9 1, 8

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 9 / 23

slide-27
SLIDE 27

Abnormal Trees

3 2 7 4 5 8 6 3, 0 3, 1 3, 1 1, 2 3, 2 1, 1 1, 0

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 10 / 23

slide-28
SLIDE 28

Abnormal Trees

3 2 7 4 5 8 6 3, 0 3, 1 3, 1 1, 2 3, 2 1, 1 1, 0 KinshipOk

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 10 / 23

slide-29
SLIDE 29

Abnormal Trees

3 2 7 4 5 8 6 3, 0 3, 1 3, 1 1, 2 3, 2 1, 1 1, 0 KinshipOk

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 10 / 23

slide-30
SLIDE 30

Abnormal Trees

3 2 7 4 5 8 6 3, 0 3, 1 3, 1 1, 2 3, 2 1, 1 1, 0 KinshipOk Abnormal root

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 10 / 23

slide-31
SLIDE 31

Abnormal Trees

3 2 7 4 5 8 6 3, 0 3, 1 3, 1 1, 2 3, 2 1, 1 1, 0 Abnormal root

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 10 / 23

slide-32
SLIDE 32

Abnormal Trees

T1 T2 T3 3 2 7 4 5 8 6 3, 0 3, 1 3, 1 1, 2 3, 2 1, 1 1, 0

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 10 / 23

slide-33
SLIDE 33

Abnormal Trees

T1 T2 T3 3 2 7 4 5 8 6 3, 0 3, 1 3, 1 1, 2 3, 2 1, 1 1, 0

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 10 / 23

slide-34
SLIDE 34

Abnormal Trees

T1 T2 T3 3 2 7 4 5 8 6 3, 0 3, 1 3, 1 1, 2 3, 2 1, 1 1, 0

Key: idR, level

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 10 / 23

slide-35
SLIDE 35

Cleaning

C

Key: idR, level Clean EBroadcast EFeedback

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 11 / 23

slide-36
SLIDE 36

Cleaning

C EB-action 6 2 8 1, 0 1, 1 1, 1

Key: idR, level Clean EBroadcast EFeedback

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 11 / 23

slide-37
SLIDE 37

Cleaning

C EB-action 6 2 8 1, 0 1, 1 1, 1

Key: idR, level Clean EBroadcast EFeedback

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 11 / 23

slide-38
SLIDE 38

Cleaning

EB

Key: idR, level Clean EBroadcast EFeedback

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 11 / 23

slide-39
SLIDE 39

Cleaning

EB EF-action 7 3 1, 5 1, 6

Key: idR, level Clean EBroadcast EFeedback

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 11 / 23

slide-40
SLIDE 40

Cleaning

EB EF-action 7 3 1, 5 1, 6

Key: idR, level Clean EBroadcast EFeedback

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 11 / 23

slide-41
SLIDE 41

Cleaning

EF

Key: idR, level Clean EBroadcast EFeedback

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 11 / 23

slide-42
SLIDE 42

Cleaning

EF R-action 6 2 8 1, 0 1, 1 1, 1

Key: idR, level Clean EBroadcast EFeedback

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 11 / 23

slide-43
SLIDE 43

Cleaning

R-action EF EF 6 2 8 6, 0 1, 1 1, 1

Key: idR, level Clean EBroadcast EFeedback

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 11 / 23

slide-44
SLIDE 44

Stabilization Time in Rounds

No alive abnormal tree created Height of an abnormal tree: at most n

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 12 / 23

slide-45
SLIDE 45

Stabilization Time in Rounds

No alive abnormal tree created Height of an abnormal tree: at most n Cleaning:

◮ EB-wave : n ◮ EF-wave : n ◮ R-wave : n Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 12 / 23

slide-46
SLIDE 46

Stabilization Time in Rounds

No alive abnormal tree created Height of an abnormal tree: at most n Cleaning:

◮ EB-wave : n ◮ EF-wave : n ◮ R-wave : n

Building of the Spanning Tree: D

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 12 / 23

slide-47
SLIDE 47

Stabilization Time in Rounds

No alive abnormal tree created Height of an abnormal tree: at most n Cleaning:

◮ EB-wave : n ◮ EF-wave : n ◮ R-wave : n

Building of the Spanning Tree: D

O(3n + D) rounds

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 12 / 23

slide-48
SLIDE 48

Lower Bound on the Worst Case Stabilization Time in Rounds

k links j = k + 3 D = n−k

2 1 n

. . .

j

. . .

5 4 3

0, 0 0, n-1 0, n-2 0, j-2 0, 3 0, 2 0, 1

Key: idR, level Clean EBroadcast EFeedback

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 13 / 23

slide-49
SLIDE 49

Lower Bound on the Worst Case Stabilization Time in Rounds

k links j = k + 3 D = n−k

EB-wave 2 1 n

. . .

j

. . .

5 4 3

0, 0 0, n-1 0, n-2 0, j-2 0, 3 0, 2 0, 1

Key: idR, level Clean EBroadcast EFeedback

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 13 / 23

slide-50
SLIDE 50

Lower Bound on the Worst Case Stabilization Time in Rounds

k links j = k + 3 D = n−k

2 1 n

. . .

j

. . .

5 4 3

0, 0 0, n-1 0, n-2 0, j-2 0, 3 0, 2 0, 1

Key: idR, level Clean EBroadcast EFeedback

n

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 13 / 23

slide-51
SLIDE 51

Lower Bound on the Worst Case Stabilization Time in Rounds

k links j = k + 3 D = n−k

EF-wave 2 1 n

. . .

j

. . .

5 4 3

0, 0 0, n-1 0, n-2 0, j-2 0, 3 0, 2 0, 1

Key: idR, level Clean EBroadcast EFeedback

n

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 13 / 23

slide-52
SLIDE 52

Lower Bound on the Worst Case Stabilization Time in Rounds

k links j = k + 3 D = n−k

2 1 n

. . .

j

. . .

5 4 3

0, 0 0, n-1 0, n-2 0, j-2 0, 3 0, 2 0, 1

Key: idR, level Clean EBroadcast EFeedback

n +n

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 13 / 23

slide-53
SLIDE 53

Lower Bound on the Worst Case Stabilization Time in Rounds

k links j = k + 3 D = n−k

R-wave 2 1 n

. . .

j

. . .

5 4 3

0, 0 0, n-1 0, n-2 0, j-2 0, 3 0, 2 0, 1

Key: idR, level Clean EBroadcast EFeedback

n +n

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 13 / 23

slide-54
SLIDE 54

Lower Bound on the Worst Case Stabilization Time in Rounds

k links j = k + 3 D = n−k

2 1 n

. . .

j

. . .

5 4 3

2, 0 1, 0 2, n-k 2, 1 2, 1 2, 1 2, 1

Key: idR, level Clean EBroadcast EFeedback

n +n +n

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 13 / 23

slide-55
SLIDE 55

Lower Bound on the Worst Case Stabilization Time in Rounds

k links j = k + 3 D = n−k

Building 2 1 n

. . .

j

. . .

5 4 3

2, 0 1, 0 2, n-k 2, 1 2, 1 2, 1 2, 1

Key: idR, level Clean EBroadcast EFeedback

n +n +n

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 13 / 23

slide-56
SLIDE 56

Lower Bound on the Worst Case Stabilization Time in Rounds

k links j = k + 3 D = n−k

2 1 n

. . .

j

. . .

5 4 3

1, n-k-1 1, 0 1, 1 1, n-k-2 1, n-k 1, n-k 1, n-k

Key: idR, level Clean EBroadcast EFeedback

n +n +n +(n − k)

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 13 / 23

slide-57
SLIDE 57

Lower Bound on the Worst Case Stabilization Time in Rounds

k links j = k + 3 D = n−k

2 1 n

. . .

j

. . .

5 4 3

1, n-k-1 1, 0 1, 1 1, n-k-2 1, n-k 1, n-k 1, n-k

Key: idR, level Clean EBroadcast EFeedback

n +n +n +(n − k)

= exactly 3n + D rounds

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 13 / 23

slide-58
SLIDE 58

Stabilization Time in Steps

Death of an abnormal tree A segment Another segment

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 14 / 23

slide-59
SLIDE 59

Stabilization Time in Steps

Death of an abnormal tree A segment Another segment

At most n alive abnormal trees + No alive abnormal tree created

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 14 / 23

slide-60
SLIDE 60

Stabilization Time in Steps

Death of an abnormal tree A segment Another segment

At most n alive abnormal trees + No alive abnormal tree created − → At most n + 1 segments

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 14 / 23

slide-61
SLIDE 61

Stabilization Time in Steps

Death of an abnormal tree A segment Another segment

At most n alive abnormal trees + No alive abnormal tree created − → At most n + 1 segments

In a segment

idR : 7 5 3 2 7 3

J-action J-action J-action EB-action EF-action R-action J-action

Death of an abnormal tree = End of the segment

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 14 / 23

slide-62
SLIDE 62

Stabilization Time in Steps

Death of an abnormal tree A segment Another segment

At most n alive abnormal trees + No alive abnormal tree created − → At most n + 1 segments

In a segment

idR : 7 5 3 2 7 3

J-action J-action J-action EB-action EF-action R-action J-action

Death of an abnormal tree = End of the segment

  • n − 1 J-actions
  • 1 EB-action
  • 1 EF-action
  • 1 R-action

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 14 / 23

slide-63
SLIDE 63

Stabilization Time in Steps

Death of an abnormal tree A segment Another segment

At most n alive abnormal trees + No alive abnormal tree created − → At most n + 1 segments

In a segment

idR : 7 5 3 2 7 3

J-action J-action J-action EB-action EF-action R-action J-action

Death of an abnormal tree = End of the segment

  • n − 1 J-actions
  • 1 EB-action
  • 1 EF-action
  • 1 R-action

⇒ O(n) actions per process

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 14 / 23

slide-64
SLIDE 64

Stabilization Time in Steps

Death of an abnormal tree A segment Another segment

At most n alive abnormal trees + No alive abnormal tree created − → At most n + 1 segments

In a segment

idR : 7 5 3 2 7 3

J-action J-action J-action EB-action EF-action R-action J-action

Death of an abnormal tree = End of the segment

  • n − 1 J-actions
  • 1 EB-action
  • 1 EF-action
  • 1 R-action

⇒ O(n) actions per process

O(n3) steps

Lower Bound:

n3 6 + 5 2n2 − 11 3 n + 2 steps

Upper Bound:

n3 2 + 2n2 + n 2 + 1 steps

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 14 / 23

slide-65
SLIDE 65

Lower Bound on the Worst Case Stabilization Time in Steps

Reset Build

2n 2n-1 2n-2 2n-3 2n-4

. . .

n+1 2n, 0 n-1, 0 n-2, 0 n-3, 0 n-4, 0 1, 0

Key: idR, level Clean EBroadcast EFeedback

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

slide-66
SLIDE 66

Lower Bound on the Worst Case Stabilization Time in Steps

Reset Build

2n 2n-1 2n-2 2n-3 2n-4

. . .

n+1 2n, 0 n-1, 0 n-2, 0 n-3, 0 n-4, 0 1, 0

Key: idR, level Clean EBroadcast EFeedback

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

slide-67
SLIDE 67

Lower Bound on the Worst Case Stabilization Time in Steps

Reset Build

2n 2n-1 2n-2 2n-3 2n-4

. . .

n+1 2n, 0 n-1, 0 n-2, 0 n-3, 0 n-4, 0 1, 0

Key: idR, level Clean EBroadcast EFeedback

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

slide-68
SLIDE 68

Lower Bound on the Worst Case Stabilization Time in Steps

2n 2n-1 2n-2 2n-3 2n-4

. . .

n+1 2n, 0 n-4, 3 n-4, 2 n-4, 1 n-4, 0 1, 0

Key: idR, level Clean EBroadcast EFeedback

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

slide-69
SLIDE 69

Lower Bound on the Worst Case Stabilization Time in Steps

2n 2n-1 2n-2 2n-3 2n-4

. . .

n+1 2n, 0 n-4, 3 n-4, 2 n-4, 1 n-4, 0 1, 0

Key: idR, level Clean EBroadcast EFeedback

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

slide-70
SLIDE 70

Lower Bound on the Worst Case Stabilization Time in Steps

2n 2n-1 2n-2 2n-3 2n-4

. . .

n+1 2n, 0 n-4, 3 n-4, 2 n-4, 1 2n-4, 0 1, 0

Key: idR, level Clean EBroadcast EFeedback

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

slide-71
SLIDE 71

Lower Bound on the Worst Case Stabilization Time in Steps

2n 2n-1 2n-2 2n-3 2n-4

. . .

n+1 2n, 0 n-4, 3 n-4, 2 2n-3, 0 2n-4, 0 1, 0

Key: idR, level Clean EBroadcast EFeedback

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

slide-72
SLIDE 72

Lower Bound on the Worst Case Stabilization Time in Steps

2n 2n-1 2n-2 2n-3 2n-4

. . .

n+1 2n, 0 n-4, 3 2n-2, 0 2n-3, 0 2n-4, 0 1, 0

Key: idR, level Clean EBroadcast EFeedback

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

slide-73
SLIDE 73

Lower Bound on the Worst Case Stabilization Time in Steps

2n 2n-1 2n-2 2n-3 2n-4

. . .

n+1 2n, 0 2n-1, 0 2n-2, 0 2n-3, 0 2n-4, 0 1, 0

Key: idR, level Clean EBroadcast EFeedback

Case of the reset of 2n − 4

processes: 2n − 1 2n − 2 2n − 3 2n − 4 . . . . . .

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

slide-74
SLIDE 74

Lower Bound on the Worst Case Stabilization Time in Steps

2n 2n-1 2n-2 2n-3 2n-4

. . .

n+1 2n, 0 2n-2, 1 2n-2, 0 2n-3, 0 2n-4, 0 1, 0

Key: idR, level Clean EBroadcast EFeedback

Case of the reset of 2n − 4

processes: 2n − 1 2n − 2 2n − 3 2n − 4 . . . . . .

idR = 2n-2 Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

slide-75
SLIDE 75

Lower Bound on the Worst Case Stabilization Time in Steps

2n 2n-1 2n-2 2n-3 2n-4

. . .

n+1 2n, 0 2n-2, 1 2n-3, 1 2n-3, 0 2n-4, 0 1, 0

Key: idR, level Clean EBroadcast EFeedback

Case of the reset of 2n − 4

processes: 2n − 1 2n − 2 2n − 3 2n − 4 . . . . . .

idR = 2n-2 idR = 2n-3 Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

slide-76
SLIDE 76

Lower Bound on the Worst Case Stabilization Time in Steps

2n 2n-1 2n-2 2n-3 2n-4

. . .

n+1 2n, 0 2n-3, 2 2n-3, 1 2n-3, 0 2n-4, 0 1, 0

Key: idR, level Clean EBroadcast EFeedback

Case of the reset of 2n − 4

processes: 2n − 1 2n − 2 2n − 3 2n − 4 . . . . . .

idR = 2n-2 idR = 2n-3 idR = 2n-3 Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

slide-77
SLIDE 77

Lower Bound on the Worst Case Stabilization Time in Steps

2n 2n-1 2n-2 2n-3 2n-4

. . .

n+1 2n, 0 2n-3, 2 2n-3, 1 2n-4, 1 2n-4, 0 1, 0

Key: idR, level Clean EBroadcast EFeedback

Case of the reset of 2n − 4

processes: 2n − 1 2n − 2 2n − 3 2n − 4 . . . . . .

idR = 2n-2 idR = 2n-3 idR = 2n-3 idR = 2n-4 Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

slide-78
SLIDE 78

Lower Bound on the Worst Case Stabilization Time in Steps

2n 2n-1 2n-2 2n-3 2n-4

. . .

n+1 2n, 0 2n-3, 2 2n-4, 2 2n-4, 1 2n-4, 0 1, 0

Key: idR, level Clean EBroadcast EFeedback

Case of the reset of 2n − 4

processes: 2n − 1 2n − 2 2n − 3 2n − 4 . . . . . .

idR = 2n-2 idR = 2n-3 idR = 2n-3 idR = 2n-4 idR = 2n-4 Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

slide-79
SLIDE 79

Lower Bound on the Worst Case Stabilization Time in Steps

2n 2n-1 2n-2 2n-3 2n-4

. . .

n+1 2n, 0 2n-4, 3 2n-4, 2 2n-4, 1 2n-4, 0 1, 0

Key: idR, level Clean EBroadcast EFeedback

Case of the reset of 2n − 4

processes: 2n − 1 2n − 2 2n − 3 2n − 4 . . . . . .

idR = 2n-2 idR = 2n-3 idR = 2n-3 idR = 2n-4 idR = 2n-4 idR = 2n-4 Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

slide-80
SLIDE 80

Lower Bound on the Worst Case Stabilization Time in Steps

2n 2n-1 2n-2 2n-3 2n-4

. . .

n+1 2n, 0 2n-4, 3 2n-4, 2 2n-4, 1 2n-4, 0 1, 0

Key: idR, level Clean EBroadcast EFeedback

Case of the reset of 2n − 4

processes: 2n − 1 2n − 2 2n − 3 2n − 4 . . . . . .

idR = 2n-2 idR = 2n-3 idR = 2n-3 idR = 2n-4 idR = 2n-4 idR = 2n-4

j−1

i=1 i

j = 4

Θ(n) reset ⇒

n

  • j=1

j−1

  • i=1

i ⇒ Θ(n3) steps

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

slide-81
SLIDE 81

Analytical Study of Datta et al, 20113

3Datta, Larmore, and Vemula. Self-stabilizing Leader Election in Optimal Space under an Arbitrary

  • Scheduler. 2011

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 16 / 23

slide-82
SLIDE 82

Principles

Join a tree

1 4 3 2 6 1, 0 1, 1 1, 1 2, 0 6, 0

Key: idR, level Can be joined Cannot be joined

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 17 / 23

slide-83
SLIDE 83

Principles

Join a tree

1 4 3 2 6 1, 0 1, 1 1, 1 1, 2 6, 0

Key: idR, level Can be joined Cannot be joined

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 17 / 23

slide-84
SLIDE 84

Principles

Change of color

4 7 2 5 1, 2 1, 3 1, 4 1, 4 4 7 2 5 1, 2 1, 3 1, 4 1, 4

Key: idR, level Can be joined Cannot be joined

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 18 / 23

slide-85
SLIDE 85

Principles

Change of color

4 7 2 5 1, 2 1, 3 1, 4 1, 4 4 7 2 5 1, 2 1, 3 1, 4 1, 4

Key: idR, level Can be joined Cannot be joined

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 18 / 23

slide-86
SLIDE 86

Principles

Change of color

4 7 2 5 1, 2 1, 3 1, 4 1, 4 4 7 2 5 1, 2 1, 3 1, 4 1, 4

Key: idR, level Can be joined Cannot be joined

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 18 / 23

slide-87
SLIDE 87

Principles

Color Waves Absorption

Normal tree Abnormal tree

Key: idR, level Can be joined Cannot be joined

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 19 / 23

slide-88
SLIDE 88

Principles

Color Waves Absorption

Normal tree Abnormal tree

Key: idR, level Can be joined Cannot be joined

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 19 / 23

slide-89
SLIDE 89

Principles

Color Waves Absorption

Normal tree Abnormal tree

Key: idR, level Can be joined Cannot be joined

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 19 / 23

slide-90
SLIDE 90

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-91
SLIDE 91

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-92
SLIDE 92

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-93
SLIDE 93

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-94
SLIDE 94

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-95
SLIDE 95

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-96
SLIDE 96

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-97
SLIDE 97

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-98
SLIDE 98

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-99
SLIDE 99

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-100
SLIDE 100

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-101
SLIDE 101

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-102
SLIDE 102

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-103
SLIDE 103

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-104
SLIDE 104

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-105
SLIDE 105

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-106
SLIDE 106

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-107
SLIDE 107

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-108
SLIDE 108

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-109
SLIDE 109

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-110
SLIDE 110

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-111
SLIDE 111

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-112
SLIDE 112

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-113
SLIDE 113

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-114
SLIDE 114

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-115
SLIDE 115

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-116
SLIDE 116

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-117
SLIDE 117

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-118
SLIDE 118

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-119
SLIDE 119

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-120
SLIDE 120

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-121
SLIDE 121

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-122
SLIDE 122

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-123
SLIDE 123

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-124
SLIDE 124

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-125
SLIDE 125

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-126
SLIDE 126

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-127
SLIDE 127

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-128
SLIDE 128

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-129
SLIDE 129

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-130
SLIDE 130

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-131
SLIDE 131

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-132
SLIDE 132

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-133
SLIDE 133

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-134
SLIDE 134

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-135
SLIDE 135

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-136
SLIDE 136

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-137
SLIDE 137

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-138
SLIDE 138

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-139
SLIDE 139

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-140
SLIDE 140

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-141
SLIDE 141

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-142
SLIDE 142

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-143
SLIDE 143

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-144
SLIDE 144

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-145
SLIDE 145

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-146
SLIDE 146

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-147
SLIDE 147

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-148
SLIDE 148

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-149
SLIDE 149

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-150
SLIDE 150

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-151
SLIDE 151

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β2

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-152
SLIDE 152

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β3

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-153
SLIDE 153

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β3

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-154
SLIDE 154

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β3

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-155
SLIDE 155

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β3

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-156
SLIDE 156

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β3

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-157
SLIDE 157

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β3

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-158
SLIDE 158

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β3

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-159
SLIDE 159

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β3

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-160
SLIDE 160

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β3

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-161
SLIDE 161

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β3

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-162
SLIDE 162

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β3

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-163
SLIDE 163

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β3

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-164
SLIDE 164

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β3

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-165
SLIDE 165

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β3

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-166
SLIDE 166

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β3

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-167
SLIDE 167

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β3

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-168
SLIDE 168

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β3

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-169
SLIDE 169

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β3

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-170
SLIDE 170

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β3

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-171
SLIDE 171

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β4

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-172
SLIDE 172

Datta et al, 2011

Execution in Ω(n4) steps: β = n

8

β4

β = Ω(n) ⇒ Ω(n4)

Key: (i, j).ID = (i − 1)β + j (i, j).idR = 0 Can be joined Cannot be joined

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 . . . . . . . . . . . . . . . . . . . . . . . . 1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

slide-173
SLIDE 173

Datta et al, 2011

Network for Ω(n5) steps

∀α ≥ 3, ∃ networks and executions in Ω(nα+1) steps.

Nodes from Ω(n4) New nodes

1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 9,1 10,1 11,1 12,1 13,1 14,1 15,1 16,1 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 9,2 10,2 11,2 12,2 13,2 14,2 15,2 16,2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1,β 2,β 3,β 4,β 5,β 6,β 7,β 8,β 9,β 10,β 11,β 12,β 13,β 14,β 15,β 16,β

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 21 / 23

slide-174
SLIDE 174

Perspectives

Goal

Design a self-stabilizing leader election algorithm that stabilizes in O(D) rounds.

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 22 / 23

slide-175
SLIDE 175

Perspectives

Goal

Design a self-stabilizing leader election algorithm that stabilizes in O(D) rounds.

Hypotheses

Unfair daemon Memory requirement of Θ(log n) bits/process

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 22 / 23

slide-176
SLIDE 176

Perspectives

Goal

Design a self-stabilizing leader election algorithm that stabilizes in O(D) rounds.

Hypotheses

Unfair daemon Memory requirement of Θ(log n) bits/process With the knowledge of D ≥ D, (D = O(D)) :

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 22 / 23

slide-177
SLIDE 177

Perspectives

Goal

Design a self-stabilizing leader election algorithm that stabilizes in O(D) rounds.

Hypotheses

Unfair daemon Memory requirement of Θ(log n) bits/process With the knowledge of D ≥ D, (D = O(D)) : Without any global knowledge : ??

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 22 / 23

slide-178
SLIDE 178

Thank you for your attention.

Do you have any questions ?

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 23 / 23

slide-179
SLIDE 179

Rounds

1st round 2nd round Processes Time

Key: Enabled Activated Neutralized

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 24 / 23

slide-180
SLIDE 180

Experimental Results

10 15 20 25 30 35 5 10 15 20 25 Rounds D Average stabilization time Confidence interval D Average stabilization time in rounds in UDGs (n = 1000)

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 25 / 23

slide-181
SLIDE 181

Experimental Results

50 100 150 200 100 200 300 400 500 600 700 800 900 1000 Steps n Average stabilization time Confidence interval D Average stabilization time in steps in UDGs (D = 15)

Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 26 / 23