Computation of Binary Arithmetic Sum Over an Asymmetric Diamond Network

In this paper, the problem of zero-error network function computation is considered, where in a directed acyclic network, a single sink node is required to compute with zero error a function of the source messages that are separately generated by multiple source nodes. From the information-theoretic point of view, we are interested in the fundamental computing capacity, which is defined as the average number of times that the function can be computed with zero error for one use of the network. The explicit characterization of the computing capacity in general is overwhelming difficult. The best known upper bound applicable to arbitrary network topologies and arbitrary target functions is the one proved by Guang et al. in using an approach of the cut-set strong partition. This bound is tight for all previously considered network function computation problems whose computing capacities are known. In this paper, we consider the model of computing the binary arithmetic sum over an asymmetric diamond network, which is of great importance to illustrate the combinatorial nature of network function computation problem. First, we prove a corrected upper bound 1 by using a linear programming approach, which corrects an invalid bound previously claimed in the literature. Nevertheless, this upper bound cannot bring any improvement over the best known upper bound for this model, which is also equal to 1. Further, by developing a different graph coloring approach, we obtain an improved upper bound
${}\frac {1}{\log _{3} 2+\log 3-1}~(\approx 0.822)$
. We thus show that the best known upper bound proved by Guang et al. is not tight for this model which answers the open problem that whether this bound in general is tight. On the other hand, we present an explicit code construction, which implies a lower bound
${}\frac {1}{2}\log _{3}6~(\approx 0.815)$
on the computing capacity. Comparing the improved upper and lower bounds thus obtained, there exists a rough 0.007 gap between them.