Diamond channel with partially separated relays

We consider diamond channels with a general broadcast channel p(y, z|x), with outputs Z and Y at relays 1 and 2, respectively, and where the relays 1 and 2 have noiseless links of capacities R_z and R_y, respectively, to the decoder. For the case when Y and Z are deterministic functions of X, we establish the capacity. We next give an upper bound for the capacity of the class of diamond channels with a physically degraded broadcast channel, i.e., when X → Y → Z forms a Markov chain. We show that this upper bound is tight, if in addition to X → Y → Z, the output of relay 2, i.e., Y, is a deterministic function of X. We finally consider the diamond channel with partially separated relays, i.e., when the output of relay 2 is available at relay 1. We establish the capacity for this model in two cases, a) when the broadcast channel is physically degraded, i.e., when X → Y → Z forms a Markov chain, and b) when the broadcast channel is semi-deterministic, i.e, when Y = f(X). For both of these cases, we show that the capacity is equal to the cut-set bound. This final result shows that even partial feedback from the decoder to relays strictly increases the capacity of the diamond channel.