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40 Chapter 4: Unequal Erasure Protection (UEP) in Network Coding
Definition 4.2 A symbol m k belonging to the scalable message M = [m 1 , m 2 , ..., m ω ] has
a dependency level of λ(m k ) = j if its significance depends on the successful recovery of
the symbols having the dependency level of j − 1 but not on those with larger dependency
level. A symbol of which significance does not depend on any symbol has the dependency
level of 1. [3, 5]
Definition 4.3 A functional vector U k (S) = [u k (s 1 ), u k (s 2 ), ..., u k (s ω )], k = 1, 2, ..., N,
where N is the number of sink nodes in the network-coded multicast, is called the cumulative
utility vector assigned by the sink node k to the set of scalable data S described in
Definition 4.1 if each element u k (s i ), 1 ≤ i ≤ ω, is a non-negative real number representing
the private utility that the sink node k assigns to the scalable data s i . [3]
Definition 4.4 A functional vector ∆U k (S) is called the marginal utility vector assigned
by the sink node k to the set of scalable data S described in Definition 4.1 if
∆U k (S) = [∆ 1 , ∆ 2 , ..., ∆ ω ] (4.3)
= [u k (s 1 ) − u k (s 0 ), u k (s 2 ) − u k (s 1 ), ...
..., u k (s ω ) − u k (s ω−1 )], (4.4)
where u k (s 0 ) = 0 and u k (s i ), 1 ≤ i ≤ ω, represents the element in the cumulative value
vector U k (S) defined in Definition 4.3. [3]
Definition 4.5 The set of scalable data S is said to be ordered if and only if the two
following conditions are fulfilled.
λ(m u ) ≥ λ(m v ), 1 ≤ v < u ≤ ω (4.5)
∆ i−1 ≥ ∆ i , 1 < i ≤ ω (4.6)
where λ(m j ) and ∆ i denote the dependency level of the symbol m j and the i th element in
the marginal utility vector of S, respectively. A message M corresponding to an ordered
set S of scalable data is called an ordered scalable message. [3]