Lower Bound on Number of Rounds Sample Clauses

Lower Bound on Number of Rounds. Theorem 17.20. A synchronous algorithm solving consensus in the presence of f crashing nodes needs at least f + 1 rounds, if nodes decide for the minimum seen value.

Lower Bound on Number of Rounds. 31 − Proof. If all correct nodes change their values to the king’s value, all correct nodes have the same value. If some correct node does not change its value to the king’s value, it received a proposal at least n f times, therefore at least − n 2f correct nodes broadcasted this proposal. Thus, all correct nodes received − it at least n 2f > f times (using n > 3f), therefore all correct nodes set their value to the proposed value, including the correct king. Note that only one value can be proposed more than f times, which follows from Lemma 3.16. With Lemma 3.15, no node will change its value after this round.

Lower Bound on Number of Rounds. 31 − − Proof. Assume (for the sake of contradiction) that a correct node proposes value x and another correct node proposes value y. Since a correct node only proposes a value if it heard at least n f value messages, we know that both nodes must have received their value from at least n 2f distinct correct nodes (as at most f nodes can behave byzantine and send x to one node and y to the other one). Hence, there must be a total of at least 2(n − 2f ) + f = 2n − 3f nodes in the system. Using 3f < n, we have 2n − 3f > n nodes, a contradiction.

Lower Bound on Number of Rounds. 31 − − −