Self-stabilizing ( f , g )-Alliances with Safe Convergence

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

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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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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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No safety guarantee

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Ω(D)

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Are all illegitimate configurations

identically bad ?

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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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Quick convergence time

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• 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

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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

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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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Red nodes form a 1-minimal (0,1)-Alliance but not a minimal one

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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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p

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p

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p

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p

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p

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Pointer: authorization to leave

Nil

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p

Nil

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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)-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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How to evaluate Busy?

• NP∩ A < f(p)

(2,0)-Alliance

pBusy!

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• 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


• 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

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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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• Leaving should be locally sequential

• Example: (2,2)-Alliance

p

Neighbors14/11/13

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• Leaving should be locally sequential

• Example: (2,2)-Alliance

p

Neighbors

p

Neighbors14/11/13

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Pointer: authorization to leave

Nil

p

Neighbors

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Self-stabilizing ( f , g )-Alliances with Safe Convergence

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