UC San Diego

In this dissertation, the following network computing problem is considered. Source nodes in a directed acyclic network generate independent messages and a single receiver node computes a target function f of the messages. The objective is to maximize the average number of times f can be computed per network usage, i.e., the "computing capacity". The network coding problem for a single-receiver network is a special case of the network computing problem in which all of the source messages must be reproduced at the receiver. For network coding with a single receiver, routing is known to achieve the capacity by achieving the network min-cut upper bound. First we extend the definition of min-cut to the network computing problem and show that the generalized min-cut is an upper bound on the maximum achievable rate and is tight for computing (using coding) any target function in multi-edge tree networks and for computing linear target functions in any network. We also study the bound's tightness for different classes of target functions. In particular, we give a lower bound on the computing capacity in terms of the Steiner tree packing number and a different bound for symmetric functions. We also show that for certain networks and target functions, the computing capacity can be less than an arbitrarily small fraction of the min-cut bound. Next, we study the use of linear codes for network computing in single-receiver networks with various classes of target functions of the source messages. Such classes include reducible, injective, semi-injective, and linear target functions over finite fields. Computing capacity bounds are given with respect to these target function classes for network codes that use routing, linear coding, or nonlinear coding. Lastly, we consider the scenario in which a set of sources generate messages in a network over a finite field alphabet and a receiver node demands an arbitrary linear function of these messages. We formulate an algebraic test to determine whether an arbitrary network can compute linear functions using linear codes. We identify a class of linear functions that can be computed using linear codes in every network that satisfies a natural cut-based condition. Conversely, for another class of linear functions, we show that the cut- based condition does not guarantee the existence of a linear coding solution. For linear functions over the binary field, the two classes are complements of each other