Thesis

2.3. Optimal trees in case of single member compromise
Proof. B ∗ has the same elements as B has, therefore, the sum and the product of the elements of
B ∗ are the same as that of B, and so if B satisfies the constraints of the optimization problem,
then B ∗ does so too.
Now, let us assume that B ∗ is obtained from B with the bubble sort algorithm. The basic step
of this algorithm is to change two neighboring elements if they are not in the right order. Let us
suppose that bi < bi+1, and thus, the algorithm changes the order of bi and bi+1. Then, using
(2.2), we can express ∆R = R(B ∗ ) − R(B) as follows:
∆R = 1
N 2
⎛
⎝(bi+1 − 1) 2 b 2 i
=
=
1
N 2
⎛
⎝(bi − 1) 2 b 2 i+1
∏ ℓ
j=i+2 b2 j
ℓ∏
j=i+2
ℓ∏
j=i+2
b 2 j + (bi − 1) 2
b 2 j + (bi+1 − 1) 2
ℓ∏
j=i+2
ℓ∏
b 2 j
j=i+2
⎞
⎠ −
N 2
(
(bi+1 − 1) 2 b 2 i + (bi − 1) 2 − (bi − 1) 2 b 2 i+1 − (bi+1 − 1) 2)
∏ℓ j=i+2 b2j N 2
(
(bi+1 − 1) 2 (b 2 i − 1) − (bi − 1) 2 (b 2 i+1 − 1) )
= (bi − 1)(bi+1 − 1) ∏ℓ j=i+2 b2j N 2
((bi+1 − 1)(bi + 1) − (bi − 1)(bi+1 + 1))
Since bi ≥ 2 for all i, ∆R is non-negative if
bi + 1
bi − 1 ≥ bi+1 + 1
bi+1 − 1
But (2.5) must hold, since the function f(x) = x+1
x−1 is a monotone decreasing function, and by
assumption, bi < bi+1. This means, that when sorting the elements of B, we improve R(B) in
every step, and thus, R(B∗ ) ≥ R(B) must hold. ⋄
The following lemma provides a lower bound and an upper bound for the resistance to single
member compromise:
Lemma 2. Let B = (b1, b2, . . . bℓ) be a sorted branching factor vector (i.e., b1 ≥ b2 ≥ . . . ≥ bℓ).
We can give the following lower and upper bounds on R(B):
Proof. By definition
(
1 − 1
b1
) 2
≤ R(B) ≤
R = 1
N 2
⎛
⎝1 + (bℓ − 1) 2 +
=
( b1 − 1
b1
) 2
(
1 − 1
∑ℓ−1
(bi − 1) 2
i=1
b1
+ 1
N 2
⎛
⎝1 + (bℓ − 1) 2 +
) 2
ℓ∏
j=i+1
b 2 j
+ 4
3b 2 1
b 2 j
⎞
⎠
⎞
⎠
∑ℓ−1
(bi − 1) 2
i=2
ℓ∏
j=i+1
b 2 j
⎞
(2.5)
(2.6)
⎠ (2.7)
where it is used that N = b1b2 . . . bℓ. The lower bound in the lemma 3 follows directly from (2.7).
3
( ) 2
b1−1
Note that we could also derive the slightly better lower bound of + b1
1
N 2 from (2.7), however, we do not
need that in this chapter.
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