Network Functional Compression

In this thesis, we consider different aspects of the functional compression problem. In functional compression, the computation of a function (or, some functions) of sources is desired at the receiver(s). The rate region of this problem has been considered in the literature under certain restrictive assumptions. In Chapter 2 of this Thesis, we consider this problem for an arbitrary tree network and asymptotically lossless computations. In particular, for one-stage tree networks, we compute a rate-region and for an arbitrary tree network, we derive a rate lower bound based on the graph entropy. We introduce a new condition on colorings of source random variables' characteristic graphs called the coloring connectivity condition (C.C.C.). We show that unlike the condition mentioned in Doshi et al., this condition is necessary and sufficient for any achievable coding scheme based on colorings. We also show that, unlike entropy, graph entropy does not satisfy the chain rule. For one stage trees with correlated sources, and general trees with independent sources, we propose a modularized coding scheme based on graph colorings to perform arbitrarily closel