A Deterministic Polynomial--Time Algorithm for Constructing a Multicast Coding Scheme for Linear Deterministic Relay Networks

We propose a new way to construct a multicast coding scheme for linear deterministic relay networks. Our construction can be regarded as a generalization of the well-known multicast network coding scheme of Jaggi et al. to linear deterministic relay networks and is based on the notion of flow for a unicast session that was introduced by the authors in earlier work. We present randomized and deterministic polynomial--time versions of our algorithm and show that for a network with $g$ destinations, our deterministic algorithm can achieve the capacity in $\left\lceil \log(g+1)\right\rceil$ uses of the network.Comment: 12 pages, 2 figures, submitted to CISS 201

Network Coding: Exploiting Broadcast and Superposition in Wireless Networks

In this thesis we investigate improvements in efficiency of wireless communication networks, based on methods that are fundamentally different from the principles that form the basis of state-of-the-art technology. The first difference is that broadcast and superposition are exploited instead of reducing the wireless medium to a network of point-to-point links. The second difference is that the problem of transporting information through the network is not treated as a flow problem. Instead we allow for network coding to be used.\ud \ud First, we consider multicast network coding in settings where the multicast configuration changes over time. We show that for certain problem classes a universal network code can be constructed. One application is to efficiently tradeoff throughput against cost.\ud \ud Next, we deal with increasing energy efficiency by means of network coding in the presence of broadcast. It is demonstrated that for multiple unicast traffic in networks with nodes arranged on two and three dimensional rectangular lattices, network coding can reduce energy consumption by factors of four and six, respectively, compared to routing.\ud \ud Finally, we consider the use of superposition by allowing nodes to decode sums of messages. We introduce different deterministic models of wireless networks, representing various ways of handling broadcast and superposition. We provide lower and upper bounds on the transport capacity under these models. For networks with nodes arranged on a hexagonal lattice it is found that the capacity under a model exploiting both broadcast and superposition is at least 2.5 times, and no more than six times, the transport capacity under a model of point-to-point links

Design and Analysis of Low Complexity Network Coding Schemes

In classical network information theory, information packets are treated as commodities, and the nodes of the network are only allowed to duplicate and forward the packets. The new paradigm of network coding, which was introduced by Ahlswede et al., states that if the nodes are permitted to combine the information packets and forward a function of them, the throughput of the network can dramatically increase. In this dissertation we focused on the design and analysis of low complexity network coding schemes for different topologies of wired and wireless networks. In the first part we studied the routing capacity of wired networks. We provided a description of the routing capacity region in terms of a finite set of linear inequalities. We next used this result to study the routing capacity region of undirected ring networks for two multimessage scenarios. Finally, we used new network coding bounds to prove the optimality of routing schemes in these two scenarios. In the second part, we studied node-constrained line and star networks. We derived the multiple multicast capacity region of node-constrained line networks based on a low complexity binary linear coding scheme. For star networks, we examined the multiple unicast problem and offered a linear coding scheme. Then we made a connection between the network coding in a node-constrained star network and the problem of index coding with side information. In the third part, we studied the linear deterministic model of relay networks (LDRN). We focused on a unicast session and derived a simple capacity-achieving transmission scheme. We obtained our scheme by a connection to the submodular flow problem through the application of tools from matroid theory and submodular optimization theory. We also offered polynomial-time algorithms for calculating the capacity of the network and the optimal coding scheme. In the final part, we considered the multicasting problem in an LDRN and proposed a new way to construct a coding scheme. Our construction is based on the notion of flow for a unicast session in the third part of this dissertation. We presented randomized and deterministic polynomial-time versions of our algorithm