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

48 Chapter 4: Unequal Erasure Protection (UEP) in Network Coding
Fragouli, Soljanin, and Shokrollahi’s approach [14]. Figure 4.4 is first transformed into a
line graph shown in Fig. 4.5, where each node represents an edge from Fig. 4.4. Any two
nodes in Fig. 4.5 are connected if the corresponding edges in Fig. 4.4 are adjacent.
The nodes in Fig. 4.5 are grouped into five subtrees, each of which is bounded by
dashed lines, such that the members in each subtree are forced by the topology to have
the same GEK. For example, in the subtree T 1 , SA and AD must have the same GEK,
since, according to Fig. 4.4, the node A has only one incoming edge SA and thus can do
nothing but copy the received symbols and forward the copies to all outgoing edges AR 1 ,
AR 2 , AR 3 , and AD, hence the same GEK among them.
Accordingly, our problem of assigning GEKs to twenty-one edges is reduced to that
of assigning GEKs to five subtrees. The minimum subtree graph is shown in Fig. 4.6.
T 1
T 4
T 2
T 3
T 5
Figure 4.6: The minimum subtree graph derived from Fig. 4.5
In this case, the subtrees T 1 = {SA, AD, AR 1 , AR 2 , AR 3 }, T 2 = {SB, BD, BE, BR 2 },
T 3 = {SC, CE, CR 1 , CR 3 , CR 4 }, T 4 = {DF, F R 1 , F R 3 , F R 4 }, T 5 = {EG, GR 2 , GR 4 }.
The edges connecting T 1 and T 2 to T 4 , as well as T 2 and T 3 to T 5 in Fig. 4.6 imply
that the GEK of T 4 must be derived from those of T 1 and T 2 whereas that of T 5 must be
derived from those of T 2 and T 3 , i.e., the sets of vectors {f T1 , f T2 , f T4 } and {f T2 , f T3 , f T5 },
where f Tx
denotes the GEK of the subtree T x , must be linearly dependent.
In a similar manner, linear independence constraints can be represented by the sets
{f T1 , f T2 , f T3 }, {f T1 , f T3 , f T4 }, {f T1 , f T2 , f T5 }, and {f T3 , f T4 , f T5 }. The three GEKs in each set
must be linearly independent in order to ensure a full-rank system of linear equations
at each of the source and the sinks, when there is no erasure. For example, the source
S is connected to the subtrees T 1 , T 2 , and T 3 , hence the linearly independent constraint
{f T1 , f T2 , f T3 }.
The next two sections show how to solve the problem under these constraints with
different optimization objectives. In Section 4.9, the objective is equity of received data