Approximation of δ-TimelinessCarole Delporte-Gallet, LIAFA UMR 7089, Paris VII

Stéphane Devismes, VERIMAG UMR 5104, Grenoble I

Hugues Fauconnier, LIAFA UMR 7089, Paris VII

LIAFA

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Motivation

• Settings– Asynchronous message-passing distributed system

with process crashes• Problem– ∃t, ∃ P = p, …,q such that ∀t’>t every message sent

through P arrives to q in at most δ time units• Output– all correct processes eventually agree on such a path

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We want to solve this problem with a general algorithm

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Framework (1/3)• Eventual Property

– e.g., δ-timeliness of a path: ɸP(t) every message sent through ⇔P= p,…,q at time t arrives to q in at most δ time units

– PɸP ⇔∃t such that ∀t’>t, ɸP(t’) = true• PɸP is true ɸ⇔ P is eventually forever true

• PɸP is false ɸ⇔ P is infinitely often false

• Set of predicates SP

– e.g., {PɸP: P is an elementary path from p to q }– If ∃P ∈ SP : P is true in the run, then

• All correct processes eventually agree on P’ such that P’ is true in the run • e.g., problem of the previous slide

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Framework (2/3)• Approximate a property is often too hard !

– e.g., δ-timeliness of a link (p,q): Proposition: In any algorithm that approximates the δ-timeliness of a (p,q), p

must eventually sends messages at every clock tick

• Approximate (P,Q)– P,Q are predicates– P ⇒ Q– e.g, (δ-timeliness of P, Δ-timeliness of P) with Δ≥ δ– Set of pairs of predicates (P0,Q0), (P1,Q1) …

• Extraction Algorithm – If ∃Pi : Pi is true in the run, then

• All correct processes eventually agree on Qj such that Qj is true in the run

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Framework (3/3)

• Other problems:– Extraction of tree containing at least all correct

processes whose all paths from the root are Δ-timely

– Extraction of ring among all correct processes whose all links are Δ-timely

– …

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Outline

• Model

• Approximation and Extraction

• Case-Study: δ-Timeliness– Approximation for δ-Timeliness of a link– Extraction of δ-Timeliness graphs

• Conclusion

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System Settings

• Processes: – Synchronous (able to take a step at each clock tick → accurately measure the

time)– Some process crashes (correct / faulty)

• Communication: – Fully connected graph– Messages– Links:

• Reliable (no message loss)• FIFO

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Reliable Broadcast (possible because links are reliable)

• Two primitives: Broadcast(m), Deliver(m)– Validity: If a correct process invokes Broadcast(m)

then it eventually executes Deliver(m)– Agreement: If a process executes Deliver(m), then

all other correct processes eventually execute Deliver(m)

– Integrity: For every message m, every process executes Deliver(m) at most once, and only if sender(m) previously invokes Broadcast(m)

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Approximation Algorithm

• Let (P,Q) be a pair of predicates such that P⇒Q• An approximation algorithm A for process p of

predicates (P,Q) is an algorithm with a variable OutA

p such that:– If P is true, then OutA

p is eventually forever true

– If Q is false, then OutAp is false infinitely often

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Example

• An approximation algorithm A for process q of (Tpq(δ),Tpq(Δ))– If (p,q) is δ-timely, then OutA

q is eventually forever true

– If (p,q) is not (p,q) is Δ-timely, , then OutAq is false

infinitely often – Remark: If (p,q) is not δ-timely but is (p,q) is Δ-

timely, nothing to guarantee

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From local to global approximation• Proposition: If A is an approximation algorithm for p of

predicates local to p (P,Q), then for every correct process q there exists an approximation algorithm B for q of predicate (P,Q).

• Local to p = depend on p only, e.g:– Tpq(δ) is local to q– Pp is crashed is local to p

• Sketch: – Each time p modifies OutA

p , it (reliably) broadcasts the new value– Whenever a process q (included p) delivers a message from p, q

updates OutBq with the new value and, then sets OutB

q to true again

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Composition of approximations

• Proposition: If AP,Q and AP’,Q’ are approximation algorithms for correct process p of (P,Q) and for correct process q of predicates (P’,Q’), then for every correct process r there exists an approximation algorithm for r of predicate (P∧P’,Q∧Q’).

• Sketch: Similar to the previous proposition

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Extraction Algorithm (1/3)

• Let (Pi,Qi)i I ∈ be a set of pairs of predicates• Extraction algorithm – Each process p has a variable dp

– In each run where ⋁i I∈ Pi = true, ∃i0 I:∈• Eventual Agreement: ∃t: after time t, dp = i0 for all

correct process p • Validity: Qi0 is true

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Extraction Algorithm (2/3)

• Proposition: Let (Pi,Qi)i I ∈ be a set of predicates.. If (Ai)i I ∈is a set of approximation algorithms of (Pi,Qi)i I∈ , then there exists an extraction algorithm for (Pi,Qi)i I∈

• Sketch: – Acc[i] initialized to 0– p increments Acc[i] each time OutAi

p become false– p regularly sends Acc to all other process– Upon receiving an array A, a process sets Acc[i] to

max(A[i],Acc[i]) for all I– p chooses i0 such that Acc[i0] is minimum

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Extraction Algorithm (3/3)

• Proposition: Let (Pi,Qi)i I ∈ be a set of predicates. If (Ai)i I ∈ is a set of approximation algorithms of (Pi,Qi)i I∈ , then there exists an communication-efficient extraction algorithm A for (Pi,Qi)i I∈

• Communication-efficiency: In each run where ⋁i I∈ Pi = true, ∃t: after time t– ∃j I : every process ∈ p reads only OutAj

p

– No message is sent by A

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Summary

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Local approximation of predicate

Global approximation of predicate

Composition

Extraction Algorithm

Local approximation of predicate

Global approximation of predicate

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Approximation of Link δ-Timeliness (1/3)

• Proposition: Without additional assumption on the system, it is not possible to approximate (Tpq(δ) , Tpq(Δ))

• Sketch: If the links are asynchronous and the local clocks are not synchronized, then it is impossible to evaluate the time to traverse the link (p,q)

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Approximation of Link δ-Timeliness (2/3)

• Theorem: If processes are equipped of perfectly synchronized clocks, then approximation of (Tpq(δ) , Tpq(δ+K)) is possible

• Theorem: If a ∃ Γ-timely path from q to p, then approximation of (Tpq(δ) , Tpq(δ+Γ+K)) is possible

• These results can be generalized for paths(Hence we solve our initial problem)

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Approximation of Link δ-Timeliness (3/3)

• Assume processes are equipped of perfectly synchronized clocks

• Approximation algorithm of (Tpq(δ) , Tpq(δ+K)) for process q– Every K time, p send to q a message including its local

time– If a message spends more than δ time or q does not

receive message from p during δ+K, then• OutA

p ← false

– OutAp is reset to true every n time

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Approximation of crashed processes

• Approximation for every correct process q of (Pp is crashed,Pp is crashed)– If p is faulty, then OutA

q is eventually forever true – If p is correct, then OutA

q is false infinitely often

• Algorithm:– OutA

p ← false– p regularly sends messages to q,– Upon receiving a message from p:

• OutAq ← false

– OutAq is reset to true every n time

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Extraction of δ-Timeliness graphs

• I.e., a graph G(V,E) such that– All processes in ∏/V is crashed– E only contains δ-Timely links

• Easy: TGGi ≣⋀e Edges(G∈ i) Te(δ) ⋀ ⋀ v Nodes(G∉ i) Pv is crashed

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Properties of the extracted graph (1/2)

• Proposition: If Gi is the extracted graph, then (Nodes(Gi[Correct(R)]), Nodes(Gi[Faulty(R)])) is a dicut of Gi

• (X,Y) is a dicut of G=(S,E) iff (X,Y) is a partition of S such that there is no (directed) link from Y to X

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Correct Faulty

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Properties of the extracted graph (2/2)

• Proposition: If there is at least one correct process and (Gi)i I∈ only contains strongly connected graphs, then the extracted graph only contains exactly all the correct processes

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Extraction of δ-Timeliness graphs

• We can extract:– A tree containing at least all correct processes

whose all paths from the root are Δ-timely– A ring among all correct processes whose all links

are Δ-timely– …

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Conclusion

• General framework to approximate and extract properties

• Case study: δ-Timeliness

• Perspective: approximate other properties, e.g., general timeliness

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Thank you22/09/2010