with S afe C onvergence building an f g Alliance Fabienne Carrier Ajoy K Datta Stéphane Devismes Lawrence L Larmore Yvan Rivierre Co Autors Ajoy K Datta ID: 631859
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Slide1
Self-Stabilizing Algorithm with Safe Convergence building an (f,g)-Alliance
Fabienne CarrierAjoy K. DattaStéphane DevismesLawrence L. LarmoreYvan RivierreSlide2
Co-Autors
Ajoy K. Datta & Lawrence L. Larmore
Fabienne
Carrier &
Yvan
RivierreSlide3
RoadmapSelf-stabilizationSafe convergenceThe problem
ContributionAlgorithmPerspectivesSlide4
Self-Stabilization [Dijkstra,74]Slide5
Self-Stabilization [Dijkstra,74]
Transient Faults,
e.g.
, memory corruption, message lost … Slide6
Self-Stabilization [Dijkstra,74]Slide7
Self-Stabilization [Dijkstra,74]Slide8
Self-Stabilization [Dijkstra,74]Slide9
Self-Stabilization [Dijkstra,74]Slide10
Self-Stabilization [Dijkstra,74]Slide11
Self-Stabilization [Dijkstra,74]
Recover
after any number of transient faultsSlide12
Pros and ConsTolerate any finite number of transient faultsNo initialization
Large-scale networkSelf-organization in sensor networkDynamicityTopological change ≈ Transient faultTolerate only
transient faults
Eventual safety
No
stabilization detectionSlide13
Pros and ConsTolerate any finite number of transient faultsNo initialization
Large-scale networkSelf-organization in sensor networkDynamicityTopological change ≈ Transient faultTolerate only
transient faults
Eventual safety
No
stabilization detectionSlide14
Related WorkEnhancing safety:Fault-containment [Ghosh et al, PODC’96]
Superstabilization [Dolev & Herman, CJTCS’97]Time-adaptive Self-stabilization [Kutten & Patt-Shamir, PODC’97]Self-Stab + safe convergence [Kakugawa & Masuzawa, IPDPS’06]
Etc.Slide15
Back to self-stabilizationSlide16
Back to self-stabilization
No safety guaranteeSlide17
Back to self-stabilization
Ω(D)Slide18
Back to self-stabilization
Are all illegitimate configurations identically bad ?Slide19
Back to self-stabilization
Are all illegitimate configurations identically bad ?Of course, NO !Slide20
Self-stabilization + Safe Convergence
No so badReally badgoodSlide21
Self-stabilization + Safe Convergence
Quick convergence timeSlide22
Self-stabilization + Safe ConvergenceOptimal LC ⊆ feasable LCSet of feasable
LC: CLOSEDSet of optimal LC: CLOSEDQuick convergence to a feasable LC(O(1) expected)Convergence to an optimal LCSlide23
The problem: (f,g)-Alliance[Dourado et al, SSS’11]
Alliance: subset of nodesf, g: 2 functions mapping nodes to natural integersFor every process p:p ∉ Alliance ⇒ at least f(p) neighbors ∈ Alliance p ∈ Alliance ⇒ at least g(p) neighbors ∈ Alliance Slide24
Example: (f,g)-Alliance
Red nodes form a
(
1
,
0
)-AllianceSlide25
Example: (f,g)-Alliance
Red nodes
DO NOT
form a (
1
,
0
)-AllianceSlide26
(f,g)-Alliance: generalization of several problemsDominating setsK-dominating setsK-tuple
dominating setsGlobal defensive allianceGlobal offensive allianceSlide27
Minimality & 1-MinimalityLet A be a set of nodesA is a minimal (
f,g)-Alliance iff every proper subset of A is not an (f,g)-AllianceA is a 1-minimal (f,g)-Alliance iff ∀p ∈ A, A-{
p} is not an (f,g)-Alliance Slide28
Example: (0,1)-Alliance
Red nodes form
NEITHER a minimal NOR a 1-minimal
(
0
,
1
)-AllianceSlide29
Example: (0,1)-Alliance
Red nodes form
a 1-minimal
(
0
,
1
)-Alliance but not a
minimal
oneSlide30
Example: (0,1)-Alliance
Red nodes (empty set) both form a
minimal AND
a
1-minimal
(
0
,
1
)-AllianceSlide31
Property[Dourado et al, SSS’11]Every minimal (
f,g)-Alliance is a 1-minimal (f,g)-Alliance If for every node p, f(p) ≥ g(p), thenA is a minimal (f,g
)-Alliance iff A is a 1-minimal (
f
,
g
)-Alliance Slide32
ContributionSelf-Stabilizing Safe Converging Algorithm for computing the minimal (f,g)-Alliance in identified networks:
Safe ConvergenceStabilization in 4 rounds to a configuration, where an (f,g)-Alliance is definedStabilization in 4n+4 additional rounds to a configuration, where minimal (f,g)-Alliance is defined
Assumptions:If for every node p, f(
p
) ≥
g
(
p
)
and
δ(
p
) ≥
g
(
p
)
Locally shared memory model
, unfair daemon
Other complexities
Memory requirement:
O
(log
n
)
bits per process
Step complexity:
O(Δ
3n) (number of state changes) Slide33
Locally Shared Memory ModelSlide34
Locally Shared Memory ModelSlide35
Locally Shared Memory ModelSlide36
Locally Shared Memory Model
Choices of the scheduler (daemon)
Unfair daemon: no fairness assumptionSlide37
Algorithm’s main ideasSlide38
``Naïve Idea”
One
boolean
Red: ∈ A
Green: ∉ A
Two actions:
Join
Leave Slide39
``Naïve Idea”
One
boolean
Red: ∈ A
Green: ∉ A
Two actions:
Join
Leave
To obtain safe convergence, it should be harder to
leave
than to
joinSlide40
Leave the alliance p can leaves if :At least f(p) neighbors ∈ A after
p leaves ANDEach neighbor still have enough neighbors ∈ A after p leaves Slide41
At least f(p) neighbors ∈ A after p leavesLeaving should be locally sequential
Example: (2,2)-Alliance
p
NeighborsSlide42
At least f(p) neighbors ∈ A after p leavesLeaving should be locally sequential
Example: (2,2)-Alliance
p
Neighbors
p
NeighborsSlide43
At least f(p) neighbors ∈ A after p leavesLeaving should be locally sequential
Example: (2,2)-Alliance
p
Neighbors
p
NeighborsSlide44
Pointer: authorization to leaveNil
pNeighborsSlide45
Each neighbor still have enough neighbor ∈ A after p leaves A neighbor q gives an authorization only if
q still have enough neighbors ∈ A without p
p
q
(
1
,
0
)-AllianceSlide46
Each neighbor still have enough neighbor ∈ A after p leaves A neighbor q gives an authorization only if
q still have enough neighbors ∈ A without p
p
q
(
1
,
0
)-Alliance
If
q
has several choices
ID breaks tiesSlide47
Nil
Deadlock problemsNil
(
1
,
0
)-AllianceSlide48
Nil
Deadlock problemsNil
(
1
,
0
)-Alliance
Busy !Slide49
Deadlock problems
Nil
(
1
,
0
)-Alliance
Busy !
NilSlide50
Deadlock problems
Nil
(
1
,
0
)-Alliance
Busy !
Tie break !
NilSlide51
Deadlock problems
NilNil
(
1
,
0
)-Alliance
Busy !
Tie break !
NilSlide52
Deadlock problems
NilNil
(
1
,
0
)-Alliance
Busy !
Tie break !
NilSlide53
How evaluating
BusyNP∩ A < f(p) (EASY)A neighbor q of p needs that p stays in the alliance, e.g.
(
2
,
0
)-Alliance
p
q
Busy !Slide54
0
How evaluating BusyNP∩ A < f(p) (EASY)A neighbor q of p needs that p stays in the alliance, e.g.
0
3
(
2
,
0
)-Alliance
2
p
q
Busy !
1
1
NbSlide55
Last problem …(1
,0)-AllianceNil
Nil
NilSlide56
Last problem …(1
,0)-AllianceNil
Nil
Nil
Nil
Nil
NilSlide57
Last problem …(1
,0)-AllianceNil
Nil
Nil
Nil
Nil
Nil
Nil
Nil
NilSlide58
Last problem …Solution: strict alternation Nil,
Nil
Nil
NilSlide59
Last problem …Solution: strict alternation Nil,
Nil
Nil
Nil
Nil
Nil
Nil
NilSlide60
Join the Alliancep ∉ A and NP∩ A < f(p) (EASY)
A neighbor q needs that p joins the alliance: Evaluated by reading the status of q and q.NbNo neighbor points p Slide61
PerspectivesGlobal stabilization in O(D)?Self-stabilization with safe convergence without any assumption on f and g
? Slide62
Thank you!