1. Approximation of δ-Timeliness Carole Delporte-Gallet, LIAFA UMR 7089, Paris VII Stéphane Devismes, VERIMAG UMR 5104, Grenoble I Hugues Fauconnier, LIAFA UMR 7089, Paris VII LIAFA

2. 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 2SSS'201022/09/2010 We want to solve this problem with a general algorithm

3. Framework (1/3) Eventual Property – e.g., δ-timeliness of a path: ɸ P (t) ⇔ every message sent through P = p,…,q at time t arrives to q in at most δ time units – Pɸ P ⇔∃ t such that ∀ t’>t, ɸ P (t’) = true Pɸ P is true ⇔ ɸ P is eventually forever true Pɸ P is false ⇔ ɸ P is infinitely often false Set of predicates S P – e.g., {Pɸ P : P is an elementary path from p to q } – If ∃ P ∈ S P : 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 22/09/2010SSS'20103

4. 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–P ⇒ Q – e.g, (δ-timeliness of P, Δ-timeliness of P ) with Δ≥ δ – Set of pairs of predicates (P 0,Q 0 ), (P 1,Q 1 ) … Extraction Algorithm – If ∃ P i : P i is true in the run, then All correct processes eventually agree on Q j such that Q j is true in the run 22/09/2010SSS'20104

5. 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 – … 22/09/2010SSS'20105

6. Outline Model Approximation and Extraction Case-Study: δ-Timeliness – Approximation for δ-Timeliness of a link – Extraction of δ-Timeliness graphs Conclusion 6SSS'201022/09/2010

7. 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 7SSS'201022/09/2010

8. 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) 22/09/2010SSS'20108

9. 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 Out A p such that: – If P is true, then Out A p is eventually forever true – If Q is false, then Out A p is false infinitely often 22/09/2010SSS'20109

10. Example An approximation algorithm A for process q of (T pq (δ),T pq (Δ)) – If (p,q) is δ-timely, then Out A q is eventually forever true – If (p,q) is not (p,q) is Δ-timely,, then Out A q is false infinitely often – Remark: If (p,q) is not δ-timely but is (p,q) is Δ- timely, nothing to guarantee 22/09/2010SSS'201010

11. 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: – T pq (δ) is local to q – P p is crashed is local to p Sketch: – Each time p modifies Out A p, it (reliably) broadcasts the new value – Whenever a process q (included p) delivers a message from p, q updates Out B q with the new value and, then sets Out B q to true again 22/09/2010SSS'201011

12. Composition of approximations Proposition: If A P,Q and A P’,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 22/09/2010SSS'201012

13. Extraction Algorithm (1/3) Let (P i,Q i ) i ∈ I be a set of pairs of predicates Extraction algorithm – Each process p has a variable d p – In each run where ⋁ i ∈ I P i = true, ∃ i 0 ∈ I: Eventual Agreement: ∃ t: after time t, d p = i 0 for all correct process p Validity: Q i 0 is true 22/09/2010SSS'201013

14. Extraction Algorithm (2/3) Proposition: Let (P i,Q i ) i ∈ I be a set of predicates.. If (A i ) i ∈ I is a set of approximation algorithms of (P i,Q i ) i ∈ I, then there exists an extraction algorithm for (P i,Q i ) i ∈ I Sketch: – Acc[i] initialized to 0 – p increments Acc[i] each time Out Ai 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 i 0 such that Acc[i 0 ] is minimum 22/09/2010SSS'201014

15. Extraction Algorithm (3/3) Proposition: Let (P i,Q i ) i ∈ I be a set of predicates. If (A i ) i ∈ I is a set of approximation algorithms of (P i,Q i ) i ∈ I, then there exists an communication- efficient extraction algorithm A for (P i,Q i ) i ∈ I Communication-efficiency: In each run where ⋁ i ∈ I P i = true, ∃ t: after time t – ∃ j ∈ I : every process p reads only Out Aj p – No message is sent by A 22/09/2010SSS'201015

16. Summary 22/09/2010SSS'201016 Local approximation of predicate Global approximation of predicate Composition Extraction Algorithm Local approximation of predicate Global approximation of predicate

17. Approximation of Link δ-Timeliness (1/3) Proposition: Without additional assumption on the system, it is not possible to approximate (T pq (δ), T pq (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) 22/09/2010SSS'201017

18. Approximation of Link δ-Timeliness (2/3) Theorem: If processes are equipped of perfectly synchronized clocks, then approximation of (T pq (δ), T pq (δ+K)) is possible Theorem: If ∃ a Γ-timely path from q to p, then approximation of (T pq (δ), T pq (δ+Γ+K)) is possible These results can be generalized for paths (Hence we solve our initial problem) 22/09/2010SSS'201018

19. Approximation of Link δ-Timeliness (3/3) Assume processes are equipped of perfectly synchronized clocks Approximation algorithm of (T pq (δ), T pq (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 Out A p ← false – Out A p is reset to true every n time 22/09/2010SSS'201019

20. Approximation of crashed processes Approximation for every correct process q of (P p is crashed,P p is crashed ) – If p is faulty, then Out A q is eventually forever true – If p is correct, then Out A q is false infinitely often Algorithm: – Out A p ← false – p regularly sends messages to q, – Upon receiving a message from p: Out A q ← false – Out A q is reset to true every n time 22/09/2010SSS'201020

21. 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: TG G i ≣⋀ e ∈ Edges(G i ) T e (δ) ⋀ ⋀ v ∉ Nodes(G i ) P v is crashed 22/09/2010SSS'201021

22. Properties of the extracted graph (1/2) Proposition: If G i is the extracted graph, then (Nodes(G i [Correct(R)]), Nodes(G i [Faulty(R)])) is a dicut of G i (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 22/09/2010SSS'201022 Correct Faulty

23. Properties of the extracted graph (2/2) Proposition: If there is at least one correct process and (G i ) i ∈ I only contains strongly connected graphs, then the extracted graph only contains exactly all the correct processes 22/09/2010SSS'201023

24. 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 – … 22/09/2010SSS'201024

25. Conclusion General framework to approximate and extract properties Case study: δ-Timeliness Perspective: approximate other properties, e.g., general timeliness 22/09/2010SSS'201025

26. Thank you 22/09/2010SSS'201026