Consider several source nodes communicating across a wireless network to a destination node with the help of several layers of relay nodes. Recent work by has approximated the capacity of this network up to an additive gap. The communication scheme achieving this capacity approximation is based on compress-and-forward, resulting in noise accumulation as the messages traverse the network. As a consequence, the approximation gap increases linearly with the network depth. This paper develops a computation alignment strategy that can approach the capacity of a class of layered, time-varying wireless relay networks up to an approximation gap that is independent of the network depth. This strategy is based on the compute-and-forward framework, which enables relays to decode deterministic functions of the transmitted messages. Alone, compute-and-forward is insufficient to approach the capacity as it incurs a penalty for approximating the wireless channel with complex-valued coefficients by a channel with integer coefficients. Here, this penalty is circumvented by carefully matching channel realizations across time slots to create integer-valued effective channels that are well-suited to compute-and-forward. Unlike prior constant gap results, the approximation gap obtained in this paper also depends closely on the fading statistics, which are assumed to be i.i.d. Rayleigh.
For networks, channel-network separation is not always optimal: higher rates can be achieved using more sophisticated relaying techniques such as compress-and-forward (see, e.g., @cite_4 @cite_7 @cite_17 @cite_8 @cite_6 ), amplify-and-forward (see, e.g., @cite_35 @cite_19 @cite_37 @cite_30 @cite_11 ), and compute-and-forward (see, e.g., @cite_0 @cite_2 @cite_22 @cite_25 ). While for certain classes of deterministic networks the unicast and multicast capacity regions are known @cite_27 @cite_34 @cite_8 , in the general, noisy case, these problems remain open. Recent progress has been made by focusing on finding capacity approximations @cite_33 @cite_8 @cite_21 @cite_18 @cite_23 .
@cite_30 @cite_35 @cite_22 @cite_2 @cite_18 @cite_4 @cite_8 @cite_21 @cite_23 @cite_17 @cite_37 @cite_7 @cite_6 @cite_19 @cite_27 @cite_34 @cite_25 @cite_33 @cite_0 @cite_11
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A family of equivalence tools for bounding network capacities is introduced. Given a network N with node set V, the capacity of N is a set of non-negative vectors with elements corresponding to all possible multicast connections in N; a vector ℜ is in the capacity region for N if and only if it is possible to simultaneously and reliably establish all multicast connections across N at the given rates. Any other demand type with independent messages is a special case of this multiple multicast problem, and is therefore included in the given rate region. In Part I, we show that the capacity of a network N is unchanged if any independent, memoryless, point-to-point channel in N is replaced by a noiseless bit pipe with throughput equal to the removed channel's capacity. It follows that the capacity of a network comprised entirely of such point-to-point channels equals the capacity of an error-free network that replaces each channel by a noiseless bit pipe of the corresponding capacity. A related separation result was known previously for a single multicast connection over an acyclic network of independent, memoryless, point-to-point channels; our result treats general connections (e.g., a collection of simultaneous unicasts) and allows cyclic or acyclic networks. In this paper, we present an achievable rate region for double-unicast networks by assuming that the intermediate nodes perform random linear network coding, and the source and sink nodes optimize their strategies to maximize the achievable region. Such a setup can be modeled as a deterministic interference channel, whose capacity region is known. For the particular class of linear deterministic interference channels of our interest, in which the outputs and interference are linear deterministic functions of the inputs, we show that the known capacity region can be achieved by linear strategies. As a result, for a given set of network coding coefficients chosen by the intermediate nodes, the proposed linear precoding and decoding for the source and sink nodes will give the maximum achievable rate region for double-unicast networks. We further derive a suboptimal but easy-to-compute rate region that is independent of the network coding coefficients used at the intermediate nodes, and is instead specified by the min-cuts of the network. It is found that even this suboptimal region is strictly larger than the existing achievable rate regions in the literature. The capacity of multiuser networks has been a long-standing problem in information theory. Recently, have proposed a deterministic network model to approximate multiuser wireless networks. This model, known as the ADT network model, takes into account the broadcast nature of wireless medium and interference. We show that the ADT network model can be described within the algebraic network coding framework introduced by Koetter and Medard. We prove that the ADT network problem can be captured by a single matrix, and show that the min-cut of an ADT network is the rank of this matrix; thus, eliminating the need to optimize over exponential number of cuts between two nodes to compute the min-cut of an ADT network. We extend the capacity characterization for ADT networks to a more general set of connections, including single unicast multicast connection and non-multicast connections such as multiple multicast, disjoint multicast, and two-level multicast. We also provide sufficiency conditions for achievability in ADT networks for any general connection set. In addition, we show that random linear network coding, a randomized distributed algorithm for network code construction, achieves the capacity for the connections listed above. Furthermore, we extend the ADT networks to those with random erasures and cycles (thus, allowing bi-directional links). In addition, we propose an efficient linear code construction for the deterministic wireless multicast relay network model. 's proposed code construction is not guaranteed to be efficient and may potentially involve an infinite block length. Unlike several previous coding schemes, we do not attempt to find flows in the network. Instead, for a layered network, we maintain an invariant where it is required that at each stage of the code construction, certain sets of codewords are linearly independent. In this paper, we consider a discrete memoryless state-dependent relay channel with non-causal Channel State Information (CSI). We investigate three different cases in which perfect channel states can be known non-causally: i) only to the source, ii) only to the relay or iii) both to the source and to the relay node. For these three cases we establish lower bounds on the channel capacity (achievable rates) based on using Gel'fand-Pinsker coding at the nodes where the CSI is available and using Compress-and-Forward (CF) strategy at the relay. Furthermore, for the general Gaussian relay channel with additive independent and identically distributed (i.i.d) states and noise, we obtain lower bounds on the capacity for the cases in which CSI is available at the source or at the relay. We also compare our derived bounds with the previously obtained results which were based on Decode-and-Forward (DF) strategy, and we show the cases in which our derived lower bounds outperform DF based bounds, and can achieve the rates close to the upper bound.