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By C Georgiou; Alex Allister Shvartsman

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We observe that (κ−1 )σ+1 > We now define a specific adversarial strategy of adversary AC used to derive our lower bounds. Let A be an iterative algorithm that solves the DoAll problem. Let pi be the number of processors remaining at the end of the ith iteration of an execution of A and let ui denote the number of tasks that 24 Synchronous Do-All with Crashes: Perfect Knowledge and Multicast remain to be done at the end of iteration i. Initially, p0 = p and u0 = n. The adversarial strategy is defined assuming the same initial number of tasks and processors, that is, p0 = n0 .

This is done by separately assessing the cost of tolerating failures and the cost of achieving perfect knowledge. The first analysis is derived from the results obtained under the assumption of perfect knowledge (previous section). The latter is derived from the structure of the algorithm. Algorithm AN uses a multiple-coordinator approach to solve DoAll AC (n, p, f ) on crash-prone synchronous message-passing processors (p ≤ n). The model assumes that messages incur a known bounded delay and that reliable multicast is available: when a processor multicasts a message to a collection of processors, either all messages are delivered to non-faulty processors or no messages are delivered.

Thus the processor can be made aware of the restart. Byzantine processor failures. A faulty processor can behave arbitrarily. In particular, following a Byzantine failure, the processor can do nothing, do something not directed by its protocol, send arbitrary messages, or behave normally. A faulty processor controls only its own messages and its own actions, and it cannot control other processors’ messages and actions. , create and send messages that appear to have been sent by another processor).

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