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Self-stabilizing ( f , g )-Alliances with Safe Convergence. Fabienne Carrier Ajoy K. Datta Stéphane Devismes Lawrence L. Larmore Yvan Rivierre. Co- Autors. Ajoy K. Datta & Lawrence L. Larmore. Fabienne Carrier & Yvan Rivierre. Roadmap. Safe convergence - PowerPoint PPT Presentation
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Self-stabilizing (f,g)-Alliances with Safe Convergence
Fabienne CarrierAjoy K. Datta
Stéphane DevismesLawrence L. Larmore
Yvan Rivierre
SSS 2013, Osaka
Co-Autors
Ajoy K. Datta & Lawrence L. Larmore
Fabienne Carrier & Yvan Rivierre
14/11/13
SSS 2013, Osaka
Roadmap
1. Safe convergence
2. The (f,g)-alliance problem
3. Contribution
4. Algorithm
5. Perspectives
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Safe convergence
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Pros and Cons of Self-Stabilization
• Tolerate any finite number of transient faults
• No initialization– Large-scale network– Self-organization in
sensor network
• Dynamicity– Topological change ≈
Transient fault
• Tolerate only transient faults
• Eventual safety
• No stabilization detection
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Pros and Cons of Self-Stabilization
• Tolerate any finite number of transient faults
• No initialization– Large-scale network– Self-organization in
sensor network
• Dynamicity– Topological change ≈
Transient fault
• Tolerate only transient faults
• Eventual safety
• No stabilization detection
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Related Work
• Enhancing 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.
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Back to self-stabilization
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Back to self-stabilization
No safety guarantee
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Back to self-stabilization
Ω(D)
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Back to self-stabilization
Are all illegitimate configurations
identically bad ?
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Back to self-stabilization
Are all illegitimate configurations
identically bad ?Of course, NO !
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Self-stabilization + Safe Convergence
Not so bad
Really bad
good
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Self-stabilization + Safe Convergence
Quick convergence time
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Self-stabilization + Safe Convergence
• Optimal LC ⊆ feasable LC
• Set of feasable LC: CLOSED• Set of optimal LC: CLOSED
• Quick convergence to a feasable LC– (O(1) expected)
• Convergence to an optimal LC14/11/13
SSS 2013, Osaka
Self-stabilization + Safe Convergence: example [Kakugawa & Masuzawa, IPDPS’06]
• Construction of a minimal dominating set
– 1-round convergence to a dominating set • (not necessarily a minimal one)
– Then, O(D)-rounds convergence to a MINIMAL dominating set• During this phase, all configurations contain a dominating set
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The problem: (f,g)-Alliance[Dourado et al, SSS’11]
• Alliance: subset of nodes• f, g: 2 functions mapping nodes to natural
integers
• For every process p:– p ∉ Alliance at least⇒ f(p) neighbors Alliance ∈– p ∈ Alliance at least ⇒ g(p) neighbors Alliance ∈
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Example: (f,g)-Alliance
Red nodes form a (1,0)-Alliance
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Example: (f,g)-Alliance
Red nodes DO NOT form a (1,0)-Alliance
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(f,g)-Alliance: generalization of several problems
• Dominating sets• K-domination sets• K-tuple domination sets• Global defensive alliance • Global offensive alliance
E.g., Dominating set = (1,0)-alliance14/11/13
SSS 2013, Osaka
Minimality & 1-Minimality
• Let A be a set of nodes
• A is a minimal (f,g)-Alliance iff every proper subset of A is not an (f,g)-Alliance
• A is a 1-minimal (f,g)-Alliance iff ∀p ∈ A, A-{p} is not an (f,g)-Alliance
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Example: (0,1)-Alliance
Red nodes form a (0,1)-Alliance, but NEITHER a minimal NOR a 1-minimal (0,1)-Alliance
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Example: (0,1)-Alliance
Red nodes form a 1-minimal (0,1)-Alliance but not a minimal one
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Example: (0,1)-Alliance
Red nodes (empty set) both form a minimal AND a 1-minimal (0,1)-Alliance
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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), then– A is a minimal (f,g)-Alliance iff A is a 1-minimal
(f,g)-Alliance
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Contribution• Self-Stabilizing Safe Converging Algorithm for computing:
a minimal (f,g)-Alliance in identified networks – Safe Convergence
• Stabilization in 4 rounds to a configuration, where an (f,g)-Alliance is defined
• Stabilization 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(Δ3n)
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Algorithm’s main ideas
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``Naïve Idea”
One Boolean• Red: A∈• Green: A∉Two actions:• Join• Leave
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``Naïve Idea”
One boolean• Red: A∈• Green: A∉Two actions:• Join• Leave
To obtain safe convergence, it should be harder to leave than
to join
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Leave the alliance
• p can leave if :
1. At least f(p) neighbors A after ∈ p leaves AND
2. Each neighbor q still have enough neighbors A ∈after p leaves • i.e., g(q) or f(q) depending whether q belongs or not to A
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At least f(p) neighbors A after ∈ p leaves
• Leaving should be locally sequential• Example: (2,1)-Alliance
p
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At least f(p) neighbors A after ∈ p leaves
• Leaving should be locally sequential• Example: (2,1)-Alliance
p
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p
SSS 2013, Osaka
At least f(p) neighbors A after ∈ p leaves
• Leaving should be locally sequential• Example: (2,1)-Alliance
p
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At least f(p) neighbors A after ∈ p leaves
• Leaving should be locally sequential• Example: (2,1)-Alliance
p
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p
SSS 2013, Osaka
Pointer: authorization to leave
Nil
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p
Nil
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Pointer: authorization to leave
Nil
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p
Nil
SSS 2013, Osaka
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
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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)-AllianceIf q has several choicesID breaks ties
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Nil
Deadlock problems
Nil
(1,0)-Alliance
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Nil
Deadlock problems
Nil
(1,0)-Alliance
Busy!
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Deadlock problems
Nil
(1,0)-Alliance
Busy!
Nil
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Deadlock problems
Nil
(1,0)-Alliance
Busy!
Tie break! Nil
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Deadlock problems
Nil
Nil(1,0)-Alliance
Busy!
Tie break! Nil
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Deadlock problems
Nil
Nil(1,0)-Alliance
Busy!
Tie break! Nil
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How to evaluate Busy?
• NP∩ A < f(p)
(2,0)-Alliance
pBusy!
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How to evaluate Busy?
• NP∩ A < f(p)• A neighbor q of p needs that p stays in the
alliance
(2,0)-Alliance
p qBusy!
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0
How to evaluate Busy?
• NP∩ A < f(p)• A neighbor q of p needs that p stays in the
alliance
23
(2,0)-Alliance
2p q
Busy!1
1 Nb14/11/13
2 2
SSS 2013, Osaka
Last problem …
• (1,0)-Alliance
NilNil Nil
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Last problem …
• (1,0)-Alliance
NilNil Nil
NilNil Nil
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Last problem …
• (1,0)-Alliance
NilNil Nil
NilNil Nil
NilNil Nil
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Last problem …
• Solution: strict alternation Nil,
NilNil Nil
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Last problem …
• Solution: strict alternation Nil,
NilNil Nil
Nil NilNil Nil
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Join the Alliance
• p A and N∉ P∩ A < f(p) (EASY)
• A neighbor q needs that p joins the alliance:– Evaluated by reading the status of q and q.Nb
• No neighbor points to p. 14/11/13
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Perspectives
• (Total) Stabilization in O(D) using O(log n) bits?
• Self-stabilization with safe convergence without any assumption on f and g?
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Thank you!14/11/13
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At least f(p) neighbors A after ∈ p leaves
• Leaving should be locally sequential
• Example: (2,2)-Alliance
p
Neighbors14/11/13
SSS 2013, Osaka
At least f(p) neighbors A after ∈ p leaves
• Leaving should be locally sequential
• Example: (2,2)-Alliance
p
Neighbors
p
Neighbors14/11/13
SSS 2013, Osaka
At least f(p) neighbors A after ∈ p leaves
• Leaving should be locally sequential
• Example: (2,2)-Alliance
p
Neighbors
p
Neighbors14/11/13
SSS 2013, Osaka
Pointer: authorization to leave
Nil
p
Neighbors
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