[/QUOTE]The best approach is to introduce one or two messages solely for the purpose of communicating horizon size. Assume for simplicity that the Gnutella network is a tree, i.e., there is only one path between any two hosts. A key observation is that the number of hosts reachable from some host with TTL N is the sum of the number of hosts reachable from all its neighbors with TTL N-1. For formally, let H[A, n, B] be the number of hosts within n hops of host A that are reachable through B. Let H’[A, n, B] be the number of hosts within n hops of host A that are not reachable through B. Then for any host A with neighbors N1...Nm,

H[A, n, Ni]=1+H’[Ni, n-1, A]

H’[A, n, Ni]=H[A, n, N1]+…+H[A, n, N(i-1)+H[A, n, N(i+1)]+…+H[A, n, Nm]

H’[A, 0, Ni]=0

Horizon estimation now works through a sort of dynamic programming algorithm. Each host maintains the horizon size per connection, i.e., H[A, n, Ni] for all n and i. Every few seconds or so, hosts calculate their H’ tables from these values, exchange them with neighbors, and use these values to update their H tables. The total horizon size reported to the user is the sum over all i of H[A, n, Ni]. This scheme is accurate, efficient, and uses almost no bandwidth.[/QUOTE]

I would like for someone to explain those equations. I dont see what they represent as in if I plugged in values for the variables how I would work it out to get an answer. ???