Network coding for resource optimization and error correction

Abstract

In the first part of this thesis, we demonstrate the benefits of network coding for optimizing the use of various network resources. We first study the problem of minimizing the power consumption for wireless multiple unicasts. A simple XOR-based coding strategy is considered for reducing the power consumption. We present a centralized polynomial time algorithm that approximately minimizes the number of transmissions for two unicast sessions. We extend it to a greedy algorithm for general problem of multiple unicasts. Previous results on network coding for low-power wireless transmissions of multiple unicasts rely on opportunistic coding or centralized optimization to reduce the power consumption. Thus we propose a distributed strategy for reducing the power consumption with wireless multiple unicasts. Our strategy attempts to increase network coding opportunities without the overhead required for centralized design or coordination. We present a polynomial time algorithm using our strategy that maximizes the expected power savings with respect to the random choice of sources and sinks on the wireless rectangular grid. We study the problem of minimum-energy multicast using network coding in mobile ad hoc networks (MANETs). The optimal subgraph can be obtained by solving a linear program every time slot, but it leads to high computational complexity. We present a low-complexity approach, network coding with periodic recomputation, which recomputes an approximate solution at fixed time intervals, and uses this solution during each time interval. We analyze the power consumption and the complexity of network with this approach. Recently, several back-pressure type optimization algorithms with network coding are presented for multiple unicasts and multicast. Such algorithms are distributed since decisions are made locally at each node based on feedback about the size of queues at the destination node of each link. We develop a back-pressure based distributed optimization framework, which can be used for optimizing over any class of network codes. Our approach is to specify the class of coding operations by a set of generalized links, and to develop optimization tools that apply to any network composed of such generalized links. In the second part of this thesis, we study the capacity of single-source single-sink noiseless networks under adversarial attack on no more than z edges. Unlike prior papers, which assume equal capacities on all links, we allow arbitrary link capacities. Results include new upper bounds, general transmission strategies, and example networks where those bounds are tight. We introduce a new method for finding upper bounds on the linear coding capacities of arbitrary networks and show that there exists networks where the capacity is 50% greater than the best rate that can be achieved with linear coding. We also demonstrate examples where, unlike the equal link capacity case, it is necessary for intermediate nodes to do coding, nonlinear error detection or error correction in order to achieve the capacity. We introduce a new strategy called and employ this strategy on a sequence of networks designed to provide increasingly complex generalizations of the cut-set bounds. The first is a two-node network with multiple feedback links. The second is a four-node acyclic network. The third is a family of 'zig-zag' networks. In the first two cases, the guess-and-forward strategy achieves the capacity. For zig-zag networks, we derive a achievable rate of guess-and-forward strategy.