Reduced Complexity Sum-Product Algorithm for Decoding Network Codes and   In-Network Function Computation
  While the capacity, feasibility and methods to obtain codes for network coding problems are well studied, the decoding procedure and complexity have not garnered much attention. In this work, we pose the decoding problem at a sink node in a network as a marginalize a product function (MPF) problem over a Boolean semiring and use the sum-product (SP) algorithm on a suitably constructed factor graph to perform iterative decoding. We use \textit{traceback} to reduce the number of operations required for SP decoding at sink node with general demands and obtain the number of operations required for decoding using SP algorithm with and without traceback. For sinks demanding all messages, we define \textit{fast decodability} of a network code and identify a sufficient condition for the same. Next, we consider the in-network function computation problem wherein the sink nodes do not demand the source messages, but are only interested in computing a function of the messages. We present an MPF formulation for function computation at the sink nodes in this setting and use the SP algorithm to obtain the value of the demanded function. The proposed method can be used for both linear and nonlinear as well as scalar and vector codes for both decoding of messages in a network coding problem and computing linear and nonlinear functions in an in-network function computation problem. 
Anindya Gupta and B. Sundar Rajan
cs.IT math.IT