Network Coding and Wireless Physical-layer ... - Jacobs University

24 Chapter 3: Introduction to Graphs and Network Coding
3.3 Introduction to Network Coding
In the paper ”Network Information Flow” published in July 2000, Ahlswede, Cai, and
Li, stated that ”Contrary to one’s intuition, our work reveals that it is in general not
optimal to regard the information to be multicast as a fluid which can simply be routed
or replicated. Rather, by employing coding at the nodes, which we refer to as network
coding, bandwidth can in general be saved [55].”
While traditional coding is performed at the transmitter, network coding is done by
routers, denoted by Ahlswede et al. as nodes. They proved in their paper that information
that was multicast from one transmitter, the source node, to a set of receivers, the sink
nodes, could achieve its maximum flow (throughput) by network coding, without which
this optimum might not be achieved [55].
A
1 1
B
1 1
C
F
1 1 1
G
1 1
D
E
Figure 3.5: The butterfly network with edge capacities
To understand how network coding works, consider the graph in Fig. 3.5, which is
usually called the butterfly network. The graph consists of 7 nodes where the source node
A wishes to multicast some information to the sink nodes D and E via the 9 edges with
their capacities as labeled. The unit of the capacity here is bit per unit time.
Now consider Fig. 3.6, where b 1 and b 2 are multicast to D and E. (This means both D
and E will receive both b 1 and b 2 .) We can see from Fig. 3.6(a) that D receives b 1 via the