Alignment-Based Network Coding for Two-Unicast-Z Networks

In this paper, we study the wireline two-unicast-Z communication network over directed acyclic graphs. The two-unicast-$Z$ network is a two-unicast network where the destination intending to decode the second message has a priori side information of the first message. We make three contributions in this paper. First, we describe a new linear network coding algorithm for two-unicast-Z networks over the directed acyclic graphs. Our approach includes the idea of interference alignment as one of its key ingredients. For the graphs of a bounded degree, our algorithm has linear complexity in terms of the number of vertices, and the polynomial complexity in terms of the number of edges. Second, we prove that our algorithm achieves the rate pair (1, 1) whenever it is feasible in the network. Our proof serves as an alternative, albeit restricted to two-unicast-Z networks over the directed acyclic graphs, to an earlier result of Wang et al., which studied the necessary and sufficient conditions for the feasibility of the rate pair (1, 1) in two-unicast networks. Third, we provide a new proof of the classical max-flow min-cut theorem for the directed acyclic graphs.