Computation problems, such as network coding and averaging consen- sus, have become increasingly central to the study of wireless networks. Network coding, in which intermediate terminals compute and forward functions of others’ messages, is instrumental in establishing the capacity of multicast networks. Averaging consensus, in which terminals compute the mean of others’ measurements, is a canonical building block of dis- tributed estimation over sensor networks. Both problems, however, are typically studied over graphical networks, which abstract away the broad- cast and superposition properties fundamental to wireless propagation. The performance of computation in realistic wireless environments, there- fore, remains unclear.
In this thesis, I seek after near-optimal computation strategies under realistic wireless models. For both network coding and averaging con- sensus, cooperative communications plays a key role. For network cod- ing, I consider two topologies: a single-layer network in which users may signal cooperatively, and a two-transmitter, two-receiver network aided by a dedicated relay. In the former topology, I develop a decode-and- forward scheme based on a linear decomposition of nested lattice codes. For a network having two transmitters and a single receiver, the proposed
scheme is optimal in the diversity-multiplexing tradeo↵; otherwise it pro- vides significant rate gains over existing non-cooperative approaches. In the latter topology, I show that an amplify-and-forward relay strategy is optimal almost everywhere in the degrees-of-freedom. Furthermore, for symmetric channels, amplify-and-forward achieves rates near capacity for a non-trivial set of channel gains.
For averaging consensus, I consider large networks of randomly-placed nodes. Under a path-loss wireless model, I characterize the resource de- mands of consensus with respect to three metrics: energy expended, time elapsed, and time-bandwidth product consumed. I show that existing con- sensus strategies, such as gossip algorithms, are nearly order optimal in the energy expended but strictly suboptimal in the other metrics. I propose a new consensus strategy, tailored to the wireless medium and cooperative in nature, termed hierarchical averaging. Hierarchical averaging is nearly order optimal in all three metrics for a wide range of path-loss exponents. Finally, I examine consensus under a simple quantization model, show- ing that hierarchical averaging achieves a nearly order-optimal tradeo↵ between resource consumption and estimation accuracy
