Computing Vector-Linear Functions on Diamond Network

Network function computation is investigated in the letter. In the model, a target function, of which the inputs are generated at multiple source nodes, is required to be computed with zero error at a sink node over a network. Toward this end, distributed coding by integrating communication and computation in networks is regarded as an efficient solution. We are interested in its fundamental computing capacity of importance in theory and applications. In the letter, we explicitly characterize the capacities of computing all the vector-linear functions over the diamond network. The diamond network has an important topology structure which not only is typical for many multi-terminal information-theoretic problems but also illustrates the combinatorial nature of the computing problem. By applying the computing capacities thus obtained, we solve the solvability problem of vector-linear functions over the diamond network. We determine all the solvable vector-linear functions and obtain an enhanced result that the remaining vector-linear functions are not only linearly non-solvable but also non-linearly non-solvable.