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We can adapt ordering requirements from the broadcast services. Basic Multicast does not require any ordering properties. Single-Source FIFO requires that if one processor multicasts messages (possibly to different destination sets), then the messages received in each processors (if any) must be received in the same order as sent by the source. Definition of Causal Order remains the same. Instead of Total Order, which is difficult to achieve since destination sets may be different, we define another ordering property: Sub-Total Order: orders of received messages in all processors may be extended to the total order of messages; more precisely, for any messages m, m and processors pi , pj , if pi and pj receives both messages m, m then they are received in the same order by pi and pj .

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.

Snd2 .. rcvk−1 . . sndk−1 rcvk for k ≥ 2, such that snd1 is either equal to x or is executed after x by the same processor that executes x; rcvk is either equal to y or is executed before y by the same processor that executes y; rcvh is executed before sndh by the same processor, 2 ≤ h < k; and sndh sends a message that is received by rcvh+1 , 1 ≤ h < k. Note that no instruction happens before itself. We write x

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