Local Terminations and Distributed Computability in Anonymous Networks

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Local Terminations and Distributed Computability in Anonymous Networks

Jérémie Chalopin Emmanuel Godard Yves Métivier

LIF, CNRS & Aix-Marseille Université LaBRI, Université de Bordeaux

Session SHAMAN

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Distributed Computability

Question

What can becomputedby an arbitrary network, withpartial knowledgeabout its topology?

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Distributed Computability

Question

For distributed networks, computability is known to be restricted by

◮ Is it a ring, a tree? ...

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Distributed Computability

Question

◮ Are there loss of messages, or no? ...

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Distributed Computability

Question

◮ Do nodes have unique identitier? ...

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Distributed Computability

Question

◮ Is the size of the network known? ...

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Distributed Computability

Question

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Distributed Computability

Question

So the question is

Given a specification S, a family of networksF, is there a distributed algorithmAsuch that, any execution ofAon any network inF ends with final state (labels) satisfying S.

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Definitions

Is thetask(F,S)computable by a distributed algorithm onF ?

◮ F is afamilyof graphs; it represents

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Definitions

◮ F is afamilyof graphs; it represents

◮ topology: trees, rings, meshes, ... . . .

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Definitions

◮ F is afamilyof labeled graphs; it represents

◮ any information is encoded in the initial labeling:

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Definitions

◮ F is afamilyoflabeledgraphs; it represents

◮ structural information: identities, sense of direction, ...

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Definitions

◮ structural knowledge: the size, bound on the diameter, the topology ...

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Definitions

◮ thespecificationS is a graph relabelling relation

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Definitions

◮ (G,input)S(G,output): spanning tree, leader election, ...

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Definitions

◮ Is that enough to define the problem?

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Definitions

◮ Different kinds of computability, depending on the kind of terminationwe are interested in.

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Definitions

◮ Different kinds of computability, depending on the kind of terminationwe are interested in.

◮ Reliability?

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Anonymous Networks and Reliability

A network is anonymous when the initial (node) labeling is uniform.

◮ Motivations:

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Anonymous Networks and Reliability

◮ Distributed computability in a general setting,

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Anonymous Networks and Reliability

◮ Transient Faults: components failure, attacks, ...

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Anonymous Networks and Reliability

◮ Potentially Anonymous:arbitraryinitial labeling : unique identities, or “partially unique” (pseudonymous) ...

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Anonymous Networks and Reliability

◮ Communication is reliable

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Anonymous Networks and Reliability

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Anonymous Networks and Reliability

Computability in resource-constrained autonomous large scale systems

◮ Evolution of the network,

◮ Impossibility results: properties of the network that have to be“centrally” maintained.

◮ Possibility results: with efficient algorithms?

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Motivations for Local Termination

Different kinds of termination exist for distributed algorithms.

◮ Implicit termination: processes do not know that the computation is over.

◮ [Boldi and Vigna ’02] Characterization of tasks computable with implicit termination,

◮ based on fibrations and coverings,

◮ equivalence withself-stabilizing(terminating) tasks.

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Motivations for Local Termination

◮ Local termination: at some point, each process knows that it has computed its final value.

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Motivations for Local Termination

◮ Local termination: at some point, each process knows that it has computed its final value.

◮ Global termination detection: processes know that all processes have computed their final value.

◮ [C., Godard, Métivier, Tel ’07] Characterization of tasks computable with global termination detection

◮ based on coverings and quasi-coverings

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Local Terminations

◮ Local Termination:

◮ a process knows when it canstopexecuting the algorithm.

◮ [Boldi and Vigna ’99] Characterization of tasks computable with local termination

◮ based onview construction

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Local Terminations

◮ Local Termination:

◮ a process knows when it canstopexecuting the algorithm.

◮ [Boldi and Vigna ’99] Characterization of tasks computable with local termination

◮ based onview construction

◮ WeakLocal Termination:

◮ a process knows when it has computed itsfinalvalue, but can still run the algorithm (processing and forwarding other nodes messages).

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Applications

Two distributed algorithms arecomposedwhenever the second uses as inputs the outputs of the first algorithm.

◮ weak local termination: composition of distributed algorithms,

◮ local termination: garbage collection.

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Applications

Two distributed algorithms arecomposedwhenever the second uses as inputs the outputs of the first algorithm.

◮ weak local termination: composition of distributed algorithms,

◮ local termination: garbage collection.

A General Framework

Termination appears a key parameter to unify distributed computability results in communication networks.

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Older Related Works

◮ Angluin (’80)

◮ Yamashita & Kameda (’95)

◮ Boldi & Vigna (’99,’01)

◮ Mazurkiewicz (’97)

◮ Metivier et al (90’s)

◮ ...

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Model: Reliable Message Passing Systems with Transient Faults

A network is represented as a graph G with a port-numberingδ where each process can

◮ modify its state,

◮ send a message via port p,

◮ receive a message via port q.

1 1 2 1

2

1 1

2 3

3

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Model: Reliable Message Passing Systems with Transient Faults

We consider

◮ reliablecommunications,

◮ anonymoussystems.

◮ asynchronoussystems.

1 1 2 1

2

1 2 1 1

2

2 2

3

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Model: Reliable Message Passing Systems with Transient Faults

We consider

◮ reliablecommunications,

◮ (potentially)anonymoussystems.

◮ asynchronoussystems.

1 1 2 1

2

1 1

2 3

3

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Examples of Tasks with Different Terminations

Problem

◮ v ∈V is labeled by(val(v),dist(v))∈N×N∪ {∞},

◮ v ∈V should compute

max{val(u)|u is at distance at most dist(v)of v}.

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Examples of Tasks with Different Terminations

Problem

◮ If dist(v) =∞:implicit terminationis OK butweak local terminationis impossible

(2,∞) (2,∞) (2,∞)

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Examples of Tasks with Different Terminations

Problem

2 2

2

afterrrounds

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Examples of Tasks with Different Terminations

Problem

2 2

2 (2,∞) (2,∞) (2,∞) (2,∞) (5,∞)

2r afterrrounds

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Examples of Tasks with Different Terminations

Problem

2 2

2 (2,∞) (2,∞) 2 (2,∞) (5,∞)

2r afterrrounds

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Examples of Tasks with Different Terminations

Problem

◮ v ∈V is labeled by(val(v),dist(v))∈N×N,

◮ If dist(v) ∈N:weak local terminationis OK butlocal terminationis impossible

(2,1) (2,1) (2,1)

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Examples of Tasks with Different Terminations

Problem

2 2

2

afterrrounds

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Examples of Tasks with Different Terminations

Problem

2 2

2 (2,2r+2) (2,1) (2,1) (2,1) (5,1)

2r afterrrounds

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Examples of Tasks with Different Terminations

Problem

2 2

2 (2,2r+2) (2,1) 2 (2,1) (5,1)

2r afterrrounds

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Examples of Tasks with Different Terminations

Problem

2 2

2 (2,2r+2) (2,1) 2 (2,1) (5,1)

2r afterrrounds

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Examples of Tasks with Different Terminations

Problem

◮ If for neighbors u,v ,|dist(u)−dist(v)| ≤1:local terminationis OK butglobal terminationis impossible

(2,1) (2,1) (2,1)

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Examples of Tasks with Different Terminations

Problem

(2,end) (2,end) (2,end)

afterrrounds

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Examples of Tasks with Different Terminations

Problem

(2,end) (2,end)

(2,end) (2,r +2) (2,2) (5,1) (2,1) (2,1) (2,1)

2r r +1

afterrrounds

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Examples of Tasks with Different Terminations

Problem

(2,end) (2,end)

(2,end) (2,r +2) (2,2) (5,1) (2,1) (2,end) (2,1)

2r r +1

afterrrounds

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Examples of Tasks with Different Terminations

Problem

(2,end) (2,end)

(2,end) (2,r +2) (2,2) (5,1) (2,1) (2,end) (2,1)

2r r +1

afterrrounds

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Our Contribution

◮ A characterization of tasks computable withweak local termination

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Our Contribution

◮ A characterization of tasks computable with weak local termination bypolynomialalgorithms

◮ algorithms exchanging a polynomial number of bits

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Our Contribution

◮ A new characterization of tasks computable withlocal termination.

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Our Contribution

◮ A characterization of tasks computable withlocal terminationby polynomial algorithms.

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From graphs to digraphs

We represent the network by alabeleddigraph(G, δ, λ) where the state of each process is encoded by its label.

1 1 2 1

(1,1)

(2,1)

(1,2)

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Coverings

Definition

D is a covering of D^′ viaϕifϕis a locally bijective homomorphism.

D D^′

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Coverings

Definition

D is a covering of D^′ viaϕifϕis a locally bijective homomorphism.

D D^′

Lifting Lemma

If D is a covering of D^′viaϕ, in the synchronous execution of any algorithmA, v andϕ(v) behave in thesame way.

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Quasi-coverings

R v

G

v^′ K

D covering isomorphism

w

Definition: G is aquasi-coveringof D of radius R of center v .

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Quasi-coverings

R v

G

v^′ K

D covering isomorphism

w

Definition: G is aquasi-coveringof D of radius R of center v . Quasi-Lifting Lemma

In the synchronous execution of any algorithmA, v and w behave in thesame wayduringthe first R rounds.

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Necessary Condition

Suppose there exists an algorithmAthat computes(F,S).

Given any digraph G, consider the synchronous execution ofA over G.

For any v ∈V(G), let i be the first round (if it exists) where term_i(v) =TERM, then we define:

◮ result(G,v) =output_i(v),

◮ time(G,v) =i.

Otherwise, we let result(G,v) =⊥and time(G,v) =∞.

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Necessary Condition

R v

G

v^′ K

covering isomorphism

D w

◮ During the first R rounds of the execution, v and w remain in thesamestate.

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Necessary Condition

R v

G

v^′ K

D TERM w result(D,w) time(D,w)

◮ During the first R rounds of the execution, v and w remain in thesamestate.

◮ If R ≥time(D,w)or R≥time(G,w)),

result(D,w) =result(G,v)and time(D,w) =time(G,v).

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Necessary Condition

R v

G

v^′ K

D TERM w result(D,w) time(D,w)

Key Property

Two functions time and result have property(P)if for any G,D,

◮ if G is a quasi-covering of D of radius R

◮ if R≥min{time(G,v),time(D,w)}

thentime(G,v) =time(D,v)andresult(G,v) =result(D,v)

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Weak Local Termination

Theorem

A task (F,S)can be computed onF with weak local termination

⇐⇒

there exists some functionstimeandresultcomputable onF such that

◮ G S result(G)

◮ timeandresulthave Property(P)

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Weak Local Termination

Theorem

A task (F,S)can be computed onF with weak local termination by apolynomialalgorithm

⇐⇒

◮ timeandresulthave Property(P)

◮ there exists a polynomial p such that

◮ ∀G∈ F, max{time(G,v)|v ∈V(G)} ≤p(|G|)

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An Algorithm for the Sufficient Condition

A Mazurkiewicz-like AlgorithmM When executed on a graph G,

+ it reconstructs D such that G is acovering of D + it is apolynomialalgorithm

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An Algorithm for the Sufficient Condition

– it terminates onlyimplicitly

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An Algorithm for the Sufficient Condition

+ at any step i, it enables each process v to reconstruct D_i(v) such that G is aquasi-coveringof D_i(v)of center v

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An Algorithm for the Sufficient Condition

+ at any step i, it enables each process v to reconstruct D_i(v) such that G is aquasi-coveringof D_i(v)of center v – v does notnotknow theradiusof the quasi-covering

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An Algorithm for the Sufficient Condition

Szymanski, Shy and Prywes Algorithm Combined with our algorithm,

+ enables each v to compute alower boundlb(v) on the radius of the quasi-covering

+ onceMhas implicitly terminated, this lower bound tends to∞

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An Algorithm for the Sufficient Condition

Szymanski, Shy and Prywes Algorithm Combined with our algorithm,

+ enables each v to compute alower boundlb(v) on the radius of the quasi-covering

+ onceMhas implicitly terminated, this lower bound tends to∞

– we have to be careful to avoid infinite computations

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Computing the output

At each time step i, each v knows

◮ a graph D(v)such that G is a quasi-covering of D(v)of center v

◮ a vertex w(v): the image of v in D(v)

◮ a lower bound lb(v)on the radius of the quasi-covering

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Computing the output

◮ a lower bound lb(v)on the radius of the quasi-covering If lb(v)≥time(G,v), then by Property(P)

◮ time(G,v) =time(D(v),w(v))

◮ result(G,v) =result(D(v),w(v))

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Computing the output

◮ a lower bound lb(v)on the radius of the quasi-covering If lb(v)≥time(G,v), then by Property(P)

◮ time(G,v) =time(D(v),w(v))

◮ result(G,v) =result(D(v),w(v)) Problem

If lb(v)<time(G,v), result(D(v),w(v))may beundefinedif D(v) isnotinF

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Computing the output

◮ a lower bound lb(v)on the radius of the quasi-covering Solution

◮ v enumerates the graphs ofF until it finds a graph H that is a quasi-covering of D(v)of radius lb(v) of center u viaγ such thatγ(u) =w(v)

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Computing the output

◮ a lower bound lb(v)on the radius of the quasi-covering Solution

◮ v enumerates the graphs ofF until it finds a graph H that is a quasi-covering of D(v)of radius lb(v) of center u viaγ such thatγ(u) =w(v)

◮ If time(H,u)≤lb(v),

◮ output(v) :=result(H,u)

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Computing the output

lb(v) v

G

lb(v) u

H

D quasi-covering

quasi-covering

w

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Computing the output

lb(v) v

G

lb(v) u

H

time(u) result(u)

D quasi-covering

quasi-covering

w

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Computing the output

lb(v) v

G

lb(v) u

H

time(u) result(u)

D

time(w) =time(u) result(w) =result(u)

quasi-covering quasi-covering

w Sincetime(u)≤lb(v), by Property(P)

◮ time(w) =time(u)and result(w) =result(v)

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Computing the output

lb(v) v

G

lb(v) u

H

time(u) result(u)

D

time(w) =time(u) result(w) =result(u)

quasi-covering quasi-covering

w time(v)

result(v)

time(w) =time(v) result(w) =result(v)

Sincetime(u)≤lb(v), by Property(P)

◮ time(w) =time(u)and result(w) =result(v)

◮ time(v) =time(u)and result(v) =result(u)

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Complexity

Let T(G) =max{time(G,v)|v ∈V(G)}.

Proposition

Complexity of our Algorithm:

◮ O(Dn+T(G))rounds

◮ O(m²n+mT(G))messages

◮ O(∆log n+log T(G))bits per message

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Complexity

Let T(G) =max{time(G,v)|v ∈V(G)}.

Proposition

Complexity of our Algorithm:

◮ O(Dn+T(G))rounds

◮ O(m²n+mT(G))messages

◮ O(∆log n+log T(G))bits per message Corollary

If there exists an algorithmAthat computes a task(S,F) in a polynomial number of rounds(even withbigmessages), there exists apolynomialalgorithmA^′ that computes(S,F)

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Necessary Condition for Local Termination

G

u v

We define an operatorsplitsuch that

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Necessary Condition for Local Termination

G

u

v v

H=split(G,u,k)

We define an operatorsplitsuch that

◮ thebluevertices in split(G,u,k)behave like uin G during k steps

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Necessary Condition for Local Termination

G

u

v v

H=split(G,u,k)

Property

Two functions time and result have property(P^′)if for any G

◮ if k =time(G,u)

◮ then, for any v 6=u, result(G,v) =result(H,v)and time(G,v) =time(H,v)

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Local Termination

Theorem

A task (F,S)can be computed onF with local termination

⇐⇒

◮ timeandresulthave Property(P)and(P^′)

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Local Termination

Theorem

A task (F,S)can be computed onF with local termination by a polynomialalgorithm

⇐⇒

◮ timeandresulthave Property(P)and(P^′)

◮ there exists a polynomial p such that

◮ ∀G∈ F, max{time(G,v)|v ∈V(G)} ≤p(|G|)

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Conclusion

◮ Modeling arbitrary networks with reliable communications and “transient” faults.

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Conclusion

◮ Unifying questions: via termination detection,

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Conclusion

◮ Unifying results of characterizations:

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Conclusion

◮ coverings, quasi-coverings,

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Conclusion

◮ coverings, quasi-coverings,

◮ polynomial algorithms.

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Perspectives

◮ Exploiting characterization results,

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Perspectives

◮ Extensions to self-stabilizing systems (and other self-∗),

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Perspectives

◮ Others constraints:

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Perspectives

◮ Communications,

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Perspectives

◮ Communications,

◮ Interactions/scheduling,

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Perspectives

◮ Communications,

◮ Faults, dynamic networks,

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Perspectives

◮ Communications,

◮ Faults, dynamic networks,

◮ ...

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