The Local Mixing Problem

The presence of broadcast channels renders network coding particularly useful. For example, a single broadcast transmission of a proper mixture packet may simultaneously present useful information to multiple nodes. Motivated by the practical application of network coding in wireless networks, this paper formulates a local mixing problem: a source node has a set of mutually independent sources; each source is available at a subset of neighbors and needs to be transferred to another subset of neighbors; the problem is to characterize the admissible rate region, i.e., the set of channel rate allocations that can fulfill the traffic demand. This paper establishes that under the constraint that each neighbor decodes only from received symbols that are functions of sources that the node can eventually recover, the admissible rate region can be characterized by a set of linear constraints and linear mixing is optimal.