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To make D ( p ) finite, the coecient of α should be zero. i.e.
p e c e =1 .
e
Then
D ( p )=
e
p e b e +
k
p e t e .
min
t k ∈T k
(5)
e
Its dual problem is
maximize p≥ 0 D ( p ) subject to
e
p e c e =1 .
(6)
This dual problem can be resolved into independent sub-problems on different
sessions of applications with a distributed algorithm,
p ij t ij ,
minimize t k ∈T k
(7)
i
j = i
The aforementioned solution is the interactive optimization algorithm between
ISP and a P2P application, i.e. the P2P application solves the subproblem (7)
independently and delivers the optimal result t k to iTracker, after which iTracker
solves the main problem (6) to update p e .
Assumption A In the following analysis, we suppose that there exists t k
T k
that makes b e + k t e <c e ,
E , i.e. there exists feasible flow solution t k that
makes the restraint on the link capacity strictly feasible.
e
2.2 Properties of P-Distance in P4P
t e }
Theorem 1. Suppose that
is the solution
to (6) . Then there exists at least one link e whose link utilization is maximal and
its corresponding p e > 0 .The p e whose corresponding links utilization doesn't
achieve maximum is 0.
{
is the solution to (4) ,and
{
p e }
Proof: (4) is an instance of convex programming, and according to assump-
tion A, we know that the Slater constraint specification is true; hence, the strong
dual theory is true. As a result, the solution of (4) and of its dual problem (6)
satisfy the following:
b e +
k
t e
αc e ,
e
E
(8a)
1
p e c e =0
(8b)
e
p e
0 ,
e
E
(8c)
p e ( b e +
k
t e
αc e )=0 ,
e
E
(8d)
 
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