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46 Chapter 4: Unequal Erasure Protection (UEP) in Network Coding
4.7 The Problem of GEK Assignment
According to (4.8) and (4.11), the best option for the sink i is to make ρ i,j large for small
j to satisfy the UEP requirements and optimize the scalable data quality. To do so, it
first sorts the erasure probability P e,ik such that P e,ik ≤ P e,ir for any 1 ≤ k < r ≤ ω.
Then, we find that the ideal strategy is to allocate a first-UEP-level GEK to the path
with the index k = 1, such that we only need the path with the lowest erasure probability
to recover the most important prefix P 1 . In this case, ρ i,1 = 1 − P e,i1 , which is the highest
possible ρ i,1 . Next, we allocate a second-UEP level GEK to the path with next-to-thelowest
erasure probability, i.e., k = 2, such that ρ i,2 = (1 − P e,i1 ) (1 − P e,i2 ), which is the
highest possible ρ i,2 . We then keep allocating a q th level GEK to the path with the index
k = q until reaching the last path [5].
However, as earlier discussed in Section 4.3, this simple allocation scheme may not be
possible due to conflicts among sink nodes as well as linear independence and dependence
constraints of GEKs. Linear independence constraints ensure that the maximum information
flow is achieved at each node, whereas linear dependence represents topological
constrains, i.e., the GEK of any outgoing edge of the node i must be linearly dependent
on the GEKs of incoming edges [5]. These constraints will be discussed in detail in the
next section.
The conflicts among receiving nodes pose an economic problem of optimal distribution
of goods, which is discussed countlessly in economic literature. The proper solution
depends on the nature of the problem as well as the camp to which the decision maker
belongs, i.e., whether he is a socialist, capitalist, or something in between. Although we
will not give arguments about economic philosophy in general, we propose two solutions
to our GEK assignment problem according to different standards of evaluation. The first
one in Section 4.9 is more socialist and the second in Section 4.10 more capitalist.
Before presenting the two solutions in Section 4.9 and 4.10, we explain how to reduce
the complexity of the problem in the next section.