A statistical measure of tissue heterogeneity with by O'sullivan F., Roy S.

By O'sullivan F., Roy S.

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20] is needed” for solving NB-AC. It is not exactly clear what that means, but it seems to imply that at least three rounds are required in failurefree runs for deciding. The f + 2 lower bound would be therefore weak in the case of NB-AC. 6], F. 7]. This proves that even when considering the question of early deciding, Consensus and NB-AC remain two equivalent problems. 11] for Consensus. The argument is based on the following two points: (1) In an asynchronous system, a process may be so slow that it executes its first step after the other processes have decided; the decision value is the same as if the slow process had initially crashed.

ACM, 32:374–382, Apr 1985. 14 E. Gafni and L. Lamport. Disk paxos. In Int’l Symp. on DISt. Comp. (DISC), pp. 330–344, 2000. 15 R. Guerraoui. Revisiting the relationship between non-blocking atomic commitment and consensus. In Int’l Wshop on Dist. Alg. (WDAG), pp. 87–100. Sep 1995. LNCS 972. 16 R. Guerraoui. Indulgent Algorithms. In 19th ACM Symp. on Prin. of Dist. Comp. (PODC), pp. 289–297. 2000. 17 M. Hurfin and M. Raynal. A simple and fast asynchronous consensus protocol based on a weak failure detector.

Lower Bounds for Asynchronous Consensus 23 Approximate Theorem 2 If there are at least two proposers, or one malicious proposer, then there is at least a 2-message delay between the proposal of a value and the learning of that value. Approximate Theorem 3 (a) If there are at least two proposers whose proposals can be learned with a 2-message delay despite the failure of Q acceptors, or there is one such possibly malicious proposer that is not an acceptor, then N > 2Q + F + 2M . (b) If there is a single possibly malicious proposer that is also an acceptor, and whose proposals can be learned with a 2-message delay despite the failure of Q acceptors, then N > max(2Q + F + 2M − 2, Q + F + 2M ).

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