By Ivanyi A. (ed.)
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Extra info for Algorithms of informatics, vol. 2
Time complexity measures number of (synchronous) steps from the beginning to the termination. Message complexity measures the total number of pointto-point messages sent (more precisely, if a processor sends a message to three other processors in one synchronous step, it contributes three to the message complexity). The following simple algorithm completes gossip in just one synchronous step: each processor broadcasts its rumor to all processors. The algorithm is correct, because each message received contains a rumor, and a message not received means the failure of its sender.
Liveness follows from the fact that each pending triple satisfies conditions in lines 42–43 in some moment of the execution. 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.
Reliable-Causally-Ordered-Broadcast Code for any processor pi , 0 ≤ i ≤ n − 1 01 initialisation 02 T [j] ← 0 for every 0 ≤ j ≤ n − 1 03 pending list is empty 11 if bc-sendi (m, rco) occurs 12 then T [i] ← T [i] + 1 enable bc-sendi (< m, T >, rbb) 13 21 if bc-recvi (< m, Tˆ >, j, rbb) occurs then add triple (m, Tˆ, j) to pending 22 31 if (m, Tˆ, j) is the pending triple, and 32 Tˆ[j] = T [j] + 1, and 33 Tˆ[k] ≤ T [k] for every k = i 34 then enable bc-recvi (m, j, rco) remove triple (m, Tˆ, j) from pending 35 36 T [j] ← T [j] + 1 We argue that the algorithm Reliable-Causally-Ordered-Broadcast provides Reliable Causal Order broadcast service on the top of the system equipped with the Reliable Basic Broadcast service.
Algorithms of informatics, vol. 2 by Ivanyi A. (ed.)