When do only sources need to compute? On functional compression in tree networks

Abstract

In this paper, we consider the problem of functional compression for an arbitrary tree network. Suppose we have k possibly correlated source processes in a tree network, and a receiver in its root wishes to compute a deterministic function of these processes. Other nodes of this tree (called intermediate nodes) are allowed to perform some computations to satisfy the node's demand. Our objective is to find a lower bound on feasible rates for different links of this tree network (called a rate lower bound) and propose a coding scheme to achieve this rate lower bound in some cases. The rate region of functional compression problem has been an open problem. However, it has been solved for some simple networks under some special conditions. For instance, Doshi et al. (2007) considered the rate region of a network with two transmitters and a receiver under a condition on source random variables. Here, we derive a rate lower bound for an arbitrary tree network 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 the work of Doshi et al. (2007), this condition is necessary and sufficient for any achievable coding scheme. We also show that unlike the entropy, the 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 close to this rate lower bound. We show that in a general tree network case with independent sources, to achieve the rate lower bound, intermediate nodes should perform some computations. However, for a family of functions and RVs called coloring proper sets, it is sufficient to have intermediate nodes act like relays to perform arbitrarily close to the rate lower bound.