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In-Depth Information
z
∈
S
G
h
(
x
,
z
)=
∑
∑
z
2
∈
G
h
(
x
,
z
1
)+
G
h
(
x
,
z
2
)
z
1
∈
S
∩
N
S
−
N
)
Because
1
1
|
∑
y
1
∈
)
|
∑
z
1
∈
LDT
(
x
,
O
,
S
)=
G
h
(
x
,
y
1
)
−
G
h
(
x
,
z
1
)
|
|
(
h
O
h
S
O
∩
N
S
∩
N
Equation 4.10 can be rewritten as
DT
(
x
,
O
,
S
)
−
LDT
(
x
,
O
,
S
)
1
(4.11)
1
h
|
O
|
∑
y
2
∈
(
|
∑
z
2
∈
(
=
G
h
(
x
,
y
2
)
−
G
h
(
x
,
z
2
)
|
h
S
O
−
N
)
S
−
N
)
According to the definition of local neighborhood,
d
(
x
,
y
2
)
>
σ
holds for any
y
2
∈
(
O
−
N
)
, and
|
O
|>|
O
−
N
|
. Thus,
1
h
√
2
e
−
σ
2
1
|
∑
y
2
∈
(
O
−
N
)
0
<
G
h
(
x
,
y
2
)
<
2
h
2
h
|
O
π
Similarly,
d
(
x
,
z
2
)
>
σ
holds for any
z
2
∈
(
S
−
N
)
, and
|
S
|>|
S
−
N
|
. Thus,
1
1
h
√
2
e
−
σ
2
|
∑
z
2
∈
(
0
<
G
h
(
x
,
z
2
))
<
2
h
2
h
|
S
π
S
−
N
)
Therefore,
1
1
√
2
e
−
σ
2
∑
y
2
∈
(
O
−
N
)
|
∑
z
2
∈
(
S
−
N
)
1
|
G
h
(
x
,
y
2
)
−
G
h
(
x
,
z
2
)
|<
2
h
2
(4.12)
h
|
O
|
h
|
S
π
Inequality 4.9 follows from Inequality 4.11 and 4.12 immediately.
Applying Inequality 4.9 to
o
and
o
,wehave
e
−
σ
2
1
h
√
2
|
DT
(
o
,
O
,
S
)
−
LDT
(
o
,
O
,
S
)
|<
2
h
2
π
and
1
h
√
2
e
−
σ
2
|
DT
(
o
,
O
,
S
)
−
LDT
(
o
,
O
,
S
)
|<
2
h
2
π
(
Since
LDT
o
,
O
,
S
−
O
)
≥
LDT
(
o
,
O
,
S
−
O
)
,wehave
DT
(
o
,
O
,
S
)
−
DT
(
o
,
O
,
S
)= (
DT
(
o
,
O
,
S
)
−
LDT
(
o
,
O
,
S
))
−
(
DT
(
o
,
O
,
S
)
−
LDT
(
o
,
O
,
S
)) +(
LDT
(
o
,
O
,
S
)
−
LDT
(
o
,
O
,
S
))
e
−
σ
2
2
h
√
2
π
<
2
h
2
Inequality 4.8 is proved.
