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Such messages are called inquiring ones. 7. Rumor collection algorithms 631 in the next step, by way of sending reply messages. This phase converts all the collectors into disseminators. In the next phase, each disseminator sends a message to all the processors that need to be notified by it. Such messages are called notifying ones. The number of graph messages, sent by a processor at a step of the regular phase, is at most as large as the maximum node degree in the communication graph. The number of range messages, sent by a processor in a step of the regular phase, is at most as large as the number of inquiries received plus a constant - hence the global number of point-to-point range messages sent by all processors during regular phases may be accounted as a constant times the number of inquiries sent (which is one per processor per phase).

It follows that m happens before m . But if processor p0 receives m before m, which may happen, then this execution violates Causal Order. Note that trivially Single-Source FIFO requirement is preserved, since each processor broadcasts only one message. We denote by bb the Basic Broadcast service, by ssf the Single-Source FIFO, by co the Causal Order and by to the Total Order service. Reliability requirements. 6. Communication services 621 Integrity: each message m received in event bc-recv has been sent in some bc-send event.

If pi receives more than n/2 + f copies of vik (in the first round of the phase) then it sets its preference for the next phase to be vik ; otherwise it sets its preference to the phase kings preference, vkk received in the second round of the phase. 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.

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