We consider the Gaussian N-relay diamond network, where a source wants to communicate to destination node through a layer of N-relay nodes. We investigate the following question: what fraction of the capacity can we maintain using only k out of the N available relays? We show that independent of the channel configurations and operating SNR, we can always find a subset of k relays, which alone provide a rate k/(k + 1) C̅ - G, where C̅ is the information theoretic cutset upper bound on the capacity of the whole network and G is independent of the channel coefficients and the SNR and depends only on N and k (logarithmic in N and linear in k). In particular, for k = 1, this means that half of the capacity of any N-relay diamond network can be approximately achieved by routing information over a single relay. We also show that this fraction is tight: there are configurations of the N-relay diamond network, where every subset of k relays alone can at most provide approximately a fraction k/(k + 1) of the total capacity. These high-capacity k-relay subnetworks can be also discovered efficiently. We propose an algorithm that computes a constant gap approximation to the capacity of the Gaussian N-relay diamond network in O(N log N) running time and discovers a high-capacity k-relay subnetwork in O(kN) running time. This result also provides a new approximation to the capacity of the Gaussian N-relay diamond network, which is hybrid in nature: it has both multiplicative and additive gaps. In the intermediate SNR regime, this hybrid approximation is tighter than existing purely additive or purely multiplicative approximations to the capacity of this network.

We consider the N-relay Gaussian diamond network when the source and the destination have ns≥ 2 and nd≥ 2 antennas respectively. We show that when ns= nd= 2 and when the individual MISO channels from the source to each relay and the SIMO channels from each relay to the destination have the same capacity, there exists a two relay sub-network that achieves approximately all the capacity of the network. To prove this result, we establish a simple relation between the joint entropies of three Gaussian random variables, which is not implied by standard Shannon-type entropy inequalities.

We consider the Gaussian N-relay diamond network, where a source wants to communicate to a destination node through a layer of N-relay nodes. We investigate the following question: What fraction of the capacity can we maintain by using only k out of the N available relays? We show that in every Gaussian N-relay diamond network, there exists a subset of k relays which alone provide approximately k/k+1 of the total capacity. The result holds independent of the number of available relay nodes N, the channel configurations and the operating SNR. The result is tight in the sense that there exists channel configurations for N-relay diamond networks, where every subset of k relays can provide at most k/k+1 of the total capacity. The approximation is within 3 logN + 3k bits/s/Hz to the capacity. This result also provides a new approximation to the capacity of the Gaussian N-relay diamond network which is up to a multiplicative gap of 1/k+1 and additive gap of 3 logN + 3k. The current approximation results in the literature either aim to characterize the capacity within an additive gap by allowing no multiplicative gap or vice a versa. Our result suggests a new approximation approach where multiplicative and additive gaps are allowed simultaneously and are traded through an auxiliary parameter.