On the Minimum Number of Transmissions Required for Universal Recovery in Broadcast Networks
Thomas A. Courtade, Richard D. Wesel · 2010
Consider an arbitrarily connected broadcast net- work 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. In a previous paper we established necessary and sufficient conditions on the number of transmis- sions from each node required for universal recovery (in which each node recovers all k packets). However, these conditions are numerous and cumbersome. The present paper gives a series of relatively simple conditions for universal recovery that apply when the number of packets is large and the distribution of packets among the nodes is well behaved. Our first results, which apply to any fixed network topology, use only simple cuts in the network to characterize a set of transmission strategies such that for any > 0 these strategies require at most k transmissions above the minimum required for universal recovery. For certain topologies including d-regular d-connected networks, we explicitly construct transmission strate- gies that achieve universal recovery while using at most N transmissions above the minimum even when the total number of required transmissions is very large. These explicit constructions essentially resolve the problem completely for many canonical networks (e.g. cliques, rings, grids on tori, etc.).