To send a message to peer h, peer i forwards it to the neighbor whose identifier is closest to but no greater than h's identifier (modulo n).
In the example shown in Figure 2c, peer 3 is a child of peer 2 and peer 8. Thus both nodes will send a CLOSURE message to peer 3. Peer 6 remains un-connected after all phases.
Consider peer 1 and peer 2, where peer 1 is the parent of peer 2. Before peer 2 sends back the RESPONSE message to Peer 1, ideally all GREETING messages should have arrived in peer 2 so that the RESPONSE message will not be empty if any of its parents has sent a real message to peer 2 during the GREETING phase.
Relay peers in the set P(D=i-1 forward messages to peers in a set P(D=i-1 forward
Thus, we can not only more reliably deliver messages to peers but also reduce unnecessary message delivery.
Messages to peers h x m → x n, where n ∈ M m ( k ) which node m sends to all neighbors including range neighbors and SoOP neighbors .
As was shown in Table2, the complexity of BOPU is dominated by message multiplications needed by messages to peers, which scales as O ( IN | S m ( k ) | ( 1 + | M m ( k ) | + | S m ( k ) | + ∑ g ∈ G | G m ( g, k ) | ) n a 3 ).
Messages to peers h x m → x n are messages that node m sends to all neighbors, in the following form: h x m → x n ∝ h Δ m → x m ∏ s ∈ S m ( k ) h Υ s, m → x m × ∏ g ∈ G, n ∈ G m ( g, k ) h Ω n, m g → x m ∏ s ∈ S m ( k ) h Θ s, m → x m × ∏ n ′ ∈ M m ( k ) ∖ n h Γ n ′, m → x m (25).
That is, even if the peers receive messages, the peers may not forward the message to other peers.
