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Algorithms of informatics, vol. 2 by Ivanyi A. (ed.)

By Ivanyi A. (ed.)

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They are all based on the same generic code, and their efficiency depends on the quality of two data structures put in the generic algorithm. Our goal is to prove that we may find some of those data structures that obtained algorithm is always correct, and efficient if the number of crashes in the execution is at most f , where f ≤ n − 1 is a parameter. We start with description of these structures: communication graph and communication schedules. Communication graph. A graph G = (V, E) consists of a set V of vertices and a set E of edges.

After f + 1 phases, the processor decides on its preference. Each processor maintains a local array pref with n entries. We prove correctness using the following lemmas. Termination is immediate. 16 If all nonfaulty processors prefer v at the beginning of phase k, then they all prefer v at the end of phase k, for all k, 1 ≤ k ≤ f + 1. Proof Since all nonfaulty processors prefer v at the beginning of phase k, they all receive at least n − f copies of v (including their own) in the first round of phase k.

The proof of this fact is by induction on the events in the execution — suppose to the contrary that (m, t, j) is the triple with smallest (t, j) which does not satisfy conditions in lines 42–43 at any moment of the execution. It follows that there is a moment from which triple (m, t, j) has smallest (t, j) coordinates among pending triples in processor pi . Hence, starting from this moment, it must violate condition in line 43 for some k. Note that k = i, j, by updating rules in lines 23–25. It follows that processor pi never receives a message from pk with timestamp greater than t − 1, which by updating rules in lines 24-26 means that processor pk never receives a message < m, t > from j, which contradicts the liveness property of ssf broadcast service.

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