1. Systems of Distributed systems Module 2 - Distributed algorithms Teaching unit 2 – Properties of distributed algorithms Ernesto Damiani University of Bozen Lesson 5 – Formal properties

2. A lower bound Theorem –There is no algorithm that solves the consensus problem in less than f + 1 rounds in the presence of f crash failures, if n  f + 2 Prove special case f = 1 to study proof technique

3. Views Definition –Let  be an execution and let p i be a process. The view of p i in , denoted by |p i, is the subsequence of computation and message receive events that occur in p i together with the state of p i in the initial configuration of 

4. Similarity

5. Single-Failure Case Theorem –There is no algorithm that solves the consensus problem in less than 2 rounds in the presence of 1 crash failures, if n  3

6. The idea WLOG assume each process sends message to every other process Proceed by contradiction: –Consider an executionin which each process proposes 0. What is the decision value? –Consider another execution in which each process proposes 1. What is the decision value? –Show that there is a chain of admissible similar executions that relate the two executions –So what?

7. The Proof (1) Definition – i is the admissible execution of the algorithm in which  no failures occur  processes p 0, …, p i-1 propose 1

8. The Proof (2)

9. The Proof (3)

10. The executions 1 0 0 0 1 p0p0 pipi p n-1 p i+1 p i-1 1 0 0 0 1 p0p0 pipi p n-1 p i+1 p i-1 … 1 0 0 0 1 p0p0 pipi p n-1 p i+1 p i-1

11. Indistinguishibility 0 p0p0 pipi p n-1 p i+1 p i-1 1 1 1 1 1 0 0 0 0 0 0 p0p0 pipi p n-1 p i+1 p i-1 1 1 1 1 1 1 0 0 0 0

12. The Terminating Reliable Broadcast problem Termination –Every correct process eventually delivers some message Validity –If a correct process broadcasts a message m, then all correct processes eventually deliver m Agreement –If a correct process delivers a message m, then all correct processes eventually deliver m Integrity –Every correct process delivers at most one message, and, if it delivers m  SF, then some process must have broadcast m

13. Reliable Broadcast for Benign Failures Sender in round 1: 1:send m to all Process p in round k, 1  k  f+1 1:if delivered m in round k-1 and p  sender then 2:send m to all 3:halt 4:receive round k messages 5:if received m then 6:deliver(m) 7:if k = f+1 then halt 8:else if k = f+1 9:deliver(SF) 10:halt Terminates in f + 1 rounds even if failures only in round 1 even if no failures! Can we do better? find a protocol whose time complexity is proportional to t ---the number of failures that actually occurred--- rather than to f---the max number of failures that may occur What is the danger?

14. The Reliable Broadcast for benign failures (1) Terminates in f + 1 rounds –even if failures only in round 1 –even if no failures! Can we do better? –find a protocol whose time complexity is proportional to t -- -the number of failures that actually occurred--- rather than to f---the max number of failures that may occur What is the danger?

15. The Reliable Broadcast for benign failures (2)

16. Crying Wolf The danger is a dangerous chain For a dangerous chain to be a possibility at the end of round i, at least i processes must be faulty round 1 round 2 rounds 3…i-1 round i p0p0 p1p1 p2p2 p i-1 pipi p*p* p*p* p*p* p*p*

17. Early stopping: the idea What if a process p could detect how many process have failed by the end of round i ? Call faulty(p,i) the set containing the faulty processes detected by p If l faulty(p,i) l<i, there can be no active dangerous chains, and p can safely deliver SF What properties should the failure detector have to make this work? How can we implement such failure detector?

18. Early stopping: the protocol Let |faulty(p,k)| be the set of processes that have failed to send a message to p in any round 1…k 1:if p = sender then value := m else value:= ? Process p in round k, 1  k  f+1 2:send value to all 3:if value  ?then halt 4:receive round k values from all 5: |faulty(p,k)| := |faulty(p,k - 1)|U {q | p received no value from q in round k} 6:if received value v  ? then 7:value := v 8:deliver(value) 9:else if k = f+1 or |faulty(p,k)| < k then 10:value := SF 11:deliver(value) 12: if k = f+1 then halt

19. Termination (1) If in any round a process receives a value, then it delivers the value in that round If a process has only received “?” for f + 1 rounds, then it delivers SF in round f + 1

20. Termination (2) Let |faulty(p,k)| be the set of processes that have failed to send a message to p in any round 1…k 1:if p = sender then value := m else value:= ? Process p in round k, 1  k  f+1 2:send value to all 3:if value  ?then halt 4:receive round k values from all 5:|faulty(p,k)| := |faulty(p,k - 1)|U {q | p received no value from q in round k} 6:if received value v  ? then 7:value := v 8:deliver(value) 9:else if k = f+1 or |faulty(p,k)| < k then 10:value := SF 11:deliver(value) 12: if k = f+1 then halt

21. Validity (1) If the sender is correct then it sends m to all in round 1 By Validity of the underlying send and receive, every correct process will receive m by the end of round 1 By the protocol, every correct process will deliver m by the end of round 1

22. Validity (2) Let |faulty(p,k)| be the set of processes that have failed to send a message to p in any round 1…k 1:if p = sender then value := m else value:= ? Process p in round k, 1  k  f+1 2:send value to all 3:if value  ?then halt 4:receive round k values from all 5:|faulty(p,k)| := |faulty(p,k - 1)|U {q | p received no value from q in round k} 6:if received value v  ? then 7:value := v 8:deliver(value) 9:else if k = f+1 or |faulty(p,k)| < k then 10:value := SF 11:deliver(value) 12: if k = f+1 then halt FINE