The Approximate Capacity of the Gaussian N-Relay Diamond Network

We consider the Gaussian "diamond" or parallel relay network, in which a source node transmits a message to a destination node with the help of N relays. Even for the symmetric setting, in which the channel gains to the relays are identical and the channel gains from the relays are identical, the capacity of this channel is unknown in general. The best known capacity approximation is up to an additive gap of order N bits and up to a multiplicative gap of order N^2, with both gaps independent of the channel gains. In this paper, we approximate the capacity of the symmetric Gaussian N-relay diamond network up to an additive gap of 1.8 bits and up to a multiplicative gap of a factor 14. Both gaps are independent of the channel gains and, unlike the best previously known result, are also independent of the number of relays N in the network. Achievability is based on bursty amplify-and-forward, showing that this simple scheme is uniformly approximately optimal, both in the low-rate as well as in the high-rate regimes. The upper bound on capacity is based on a careful evaluation of the cut-set bound. We also present approximation results for the asymmetric Gaussian N-relay diamond network. In particular, we show that bursty amplify-and-forward combined with optimal relay selection achieves a rate within a factor O(log^4(N)) of capacity with pre-constant in the order notation independent of the channel gains.Comment: 23 pages, to appear in IEEE Transactions on Information Theor

Gaussian 1-2-1 Networks: Capacity Results for mmWave Communications

This paper proposes a new model for wireless relay networks referred to as "1-2-1 network", where two nodes can communicate only if they point "beams" at each other, while if they do not point beams at each other, no signal can be exchanged or interference can be generated. This model is motivated by millimeter wave communications where, due to the high path loss, a link between two nodes can exist only if beamforming gain at both sides is established, while in the absence of beamforming gain the signal is received well below the thermal noise floor. The main result in this paper is that the 1-2-1 network capacity can be approximated by routing information along at most $2N+2$ paths, where $N$ is the number of relays connecting a source and a destination through an arbitrary topology

Wireless Network Information Flow: A Deterministic Approach

In a wireless network with a single source and a single destination and an arbitrary number of relay nodes, what is the maximum rate of information flow achievable? We make progress on this long standing problem through a two-step approach. First we propose a deterministic channel model which captures the key wireless properties of signal strength, broadcast and superposition. We obtain an exact characterization of the capacity of a network with nodes connected by such deterministic channels. This result is a natural generalization of the celebrated max-flow min-cut theorem for wired networks. Second, we use the insights obtained from the deterministic analysis to design a new quantize-map-and-forward scheme for Gaussian networks. In this scheme, each relay quantizes the received signal at the noise level and maps it to a random Gaussian codeword for forwarding, and the final destination decodes the source's message based on the received signal. We show that, in contrast to existing schemes, this scheme can achieve the cut-set upper bound to within a gap which is independent of the channel parameters. In the case of the relay channel with a single relay as well as the two-relay Gaussian diamond network, the gap is 1 bit/s/Hz. Moreover, the scheme is universal in the sense that the relays need no knowledge of the values of the channel parameters to (approximately) achieve the rate supportable by the network. We also present extensions of the results to multicast networks, half-duplex networks and ergodic networks.Comment: To appear in IEEE transactions on Information Theory, Vol 57, No 4, April 201

Wireless Network Simplification: the Gaussian N-Relay Diamond Network

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 independent of the channel configurations and the operating SNR, we can always find a subset of k relays which alone provide a rate (kC/(k+1))-G, where C is the information theoretic cutset upper bound on the capacity of the whole network and G is a constant that 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.Comment: Submitted to Transactions on Information Theory in October 2012. The new version includes discussions on the algorithmic complexity of discovering a high-capacity subnetwork and on the performance of amplify-and-forwar

The Approximate Optimality of Simple Schedules for Half-Duplex Multi-Relay Networks

In ISIT'12 Brahma, \"{O}zg\"{u}r and Fragouli conjectured that in a half-duplex diamond relay network (a Gaussian noise network without a direct source-destination link and with $N$ non-interfering relays) an approximately optimal relay scheduling (achieving the cut-set upper bound to within a constant gap uniformly over all channel gains) exists with at most $N+1$ active states (only $N+1$ out of the $2^N$ possible relay listen-transmit configurations have a strictly positive probability). Such relay scheduling policies are said to be simple. In ITW'13 we conjectured that simple relay policies are optimal for any half-duplex Gaussian multi-relay network, that is, simple schedules are not a consequence of the diamond network's sparse topology. In this paper we formally prove the conjecture beyond Gaussian networks. In particular, for any memoryless half-duplex $N$-relay network with independent noises and for which independent inputs are approximately optimal in the cut-set upper bound, an optimal schedule exists with at most $N+1$ active states. The key step of our proof is to write the minimum of a submodular function by means of its Lov\'{a}sz extension and use the greedy algorithm for submodular polyhedra to highlight structural properties of the optimal solution. This, together with the saddle-point property of min-max problems and the existence of optimal basic feasible solutions in linear programs, proves the claim.Comment: Submitted to IEEE Information Theory Workshop (ITW) 201