Optimal exchange of packets for universal recovery in broadcast networks

Consider an arbitrarily connected broadcast network of N nodes that all wish to recover k desired packets. Each node begins with a subset of the desired packets and broadcasts messages to its neighbors. For the case where nodes must transmit an integer number of packets, this paper provides necessary and sufficient conditions which characterize the set of all transmission schemes that permit universal recovery (in which every node learns all k packets). By relaxing the integer transmission constraint, this paper gives a computable lower-bound on the amount of information required to be broadcast to achieve universal recovery. Furthermore, a network-coding-based scheme (computable in polynomial time) can always achieve this lower bound if packet splitting is permitted. In this way, packet splitting can provide a significant reduction in the amount of communication required for universal recovery. For cliques with N nodes, this paper shows that splitting the packet into N - 1 chunks allows the lower bound to be achieved with high probability.