Chapter 1 - An Introduction to Network Coding

Abstract

This chapter introduces the network coding with particular emphasis on the single-source multicasting problem. Network coding is based on a simple, yet far-reaching idea. In a packet network, rather than simply routing packets, intermediate nodes may compute and transmit functions of the packets that they receive. Many now-standard results are reviewed, in particular: the multicast capacity of a network is given by the least-capacity cut separating the source from any of its destinations; the multicast capacity is achievable with linear network coding over a finite field of sufficiently large size, and there exists a computationally efficient algorithm to find a set of capacity-achieving linear network coding coefficients; the multicast capacity is achieved with high probability over a sufficiently large field when the linear network coding coefficients are chosen at random. Allowing the nodes in a network to perform coding permits the multicast capacity of a network to be achieved with linear network coding, with potentially unbounded throughput advantage over routing. Unlike optimal routing solutions to a multicast problem, an optimal linear network coding solution can be found in polynomial time. In practice, the random linear network coding approach results in multicast protocols that are robust to changes in the underlying network topology. Despite its optimality for multicasting, it turns out that linear network coding does not, in general, suffice to achieve optimal throughputs for general flow problems.