Algebraic algorithms for network information flow

Ahlswede et al. in the seminal paper [1] have shown that in data transfer over networks, processing the data at the nodes can significantly improve the throughput. As proved by Li et al. in [2], even with a simple type of operation, namely linear operation, the throughput can still be vastly improved. In [3], it is shown that the multicasting problem over networks can be translated to an algebraic question about a polynomial associated to the network called network polynomial. In this thesis, we start from the algorithm of [3] and extend it in several directions. First, we generalize the framework of [3] to include the case where the messages can also be vectors over some fixed finite field. We also show that in contrast to the original algorithm, ours can be used to reduce the field size for the case of sending finite field elements. In both vector network code algorithm and finite field minimization, we translate the network code design problems into an algebraic problem about network polynomials. Because of the importance of the network polynomials, we investigate more properties of them and we study the relationship between these objects and the topological properties of the network. Then, we extend the result of [3] to the deterministic models for wireless relay networks, a very important class of networks that has been introduced in [4] by Avestimehr, Diggavi and Tse. Finally, for another class of networks, called broadcast networks, we introduce the transfer matrix and using its properties, we show that in the absence of public messages, processing the information at the nodes will not improve the throughput.