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54 Chapter 4: Unequal Erasure Protection (UEP) in Network Coding
then chooses the first-level GEK to allocate to T 4 . This allocation affects the marginal
values of every sink except R 1 . They are updated as follows.
Θ R 1 R 2 R 3 R 4
θ 1 123 75 85 45
θ 2 113 4 6 3
θ 3 40 2 - -
Due to the allocation of T 4 , R 3
and R 4 ’s last entries are removed since they now
have only (T 1 , T 3 ) and (T 5 , T 3 ), respectively, to bid for.
At the next announced price
25+ɛ, R 3 ’s response changes, as shown in the third row of Table 4.5.
Since R 1 is now guaranteed to win two items, it is allowed to allocate one GEK to T 3 .
After that, the marginal values are updated again, as follows.
Θ R 1 R 2 R 3 R 4
θ 1 123 71 62 42
θ 2 113 3 - -
θ 3 40 1 - -
At the price of 40, demand equals supply, as shown in the fourth row of Table 4.5, and
the market clears. Each of R 2 , R 3 , and R 4 clinches one object at this price and assigns a
GEK to T 2 , T 1 , and T 5 , respectively.
The algorithm implementing the auction at the source node is shown as follows [3].
Algorithm 4.1 Problem: Allocate a GEK f t ∈ F ω to each t ∈ T such that the sets of
linear independence and dependence constraints are satisfied.
1. Initialize the number of available items N T = |T |, the cumulative clinches C = 0,
the cumulative quantity X = 0, the individual current clinches γ i
= 0 and individual
cumulative clinches Γ i = 0 for the node i, i = 1, 2, ..., n. Set the current price ψ as the
initial price ψ 0 .
Broadcast linear dependence and independence constraints to all sink
nodes.