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

Chapter 3: Introduction to Graphs and Network Coding 25
path ABD and b 2 via ACFGD, as well as from Fig. 3.6(b) that E receives b 1 via ABFGE
and b 2 via ACE. Since the edge FG is shared between sending b 2 to D and sending b 1 to
E, its capacity for each of them is halved. Therefore, the maximum flow in this multicast
session for D, without using network codes, is 1.5 bits per unit time (1 from the path
ABD and 0.5 from ACFGD). The maximum flow for E is also the same.
A
A
b 1 b 2
b 1
b 1 b 2
b 2
B
C
B
C
b 2
F
F
b 1 b 2
b 1 b 2
G
G
b 1
D
E
D
E
(a)
(b)
Figure 3.6: Illustration of information flow within the butterfly network
However, network coding can help both D and E achieve the maximum flow of 2 bits
per unit time, as shown in Fig. 3.7. The node F does the simplest form of coding by
XORing b 1 to b 2 . As a result, in one unit time, D receives b 1 and b 1 ⊕ b 2 whereas E gets
b 2 and b 1 ⊕ b 2 , all of which can be easily decoded into b 1 and b 2 . Note that the maximum
flow of 2 bits per unit time is the same as the maximum flow for each individual flow
in the absence of another, which constitutes the limit that no multicast flow can exceed,
according to Ahlswede et al. [55].
Since real communication networks are more complicated than the butterfly network,
we need a mathematical model to treat the general problems of network coding. This is
explained in the following section.