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n
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Sak
ai
[
1
0
]
o
b
tain
e
d
th
e
d
is
tan
ce
two
lab
ellin
g
o
f
ch
o
r
d
al
g
r
ap
h
s
.
Sm
ith
a
an
d
T
h
ir
u
s
an
g
u
[
1
1
]
p
r
o
v
ed
th
e
r
e
s
u
lts
f
o
r
th
e
q
u
ad
r
ilater
al
s
n
ak
e
as 8
an
d
f
o
r
th
e
alter
n
ate
q
u
ad
r
ilater
al
s
n
ak
e
g
r
ap
h
as
5
f
o
r
≥
2
.
Ku
ju
r
e
t
a
l
.
[
1
2
]
p
r
o
v
e
d
th
at
2
,
1
(
,
)
≤
13
,
wh
er
e
th
e
b
lo
o
m
g
r
a
p
h
is
,
(
,
>
2
)
.
Fu
r
th
er
m
o
r
e,
Ye
n
o
k
e
et
a
l.
[
1
3
]
f
o
u
n
d
th
e
b
o
u
n
d
s
f
o
r
s
ilicate
an
d
o
x
id
e
n
etwo
r
k
s
as
2
,
1
(
(
)
)
≤
8
an
d
2
,
1
(
(
)
)
≤
10
,
r
esp
ec
tiv
ely
.
Fo
r
a
co
n
n
ec
ted
g
r
ap
h
A,
r
ad
i
o
lab
ellin
g
is
an
in
jectio
n
,
k
,
f
r
o
m
th
e
v
er
tices
o
f
to
s
u
ch
th
at
d
r
ep
r
esen
ts
th
e
d
iam
eter
o
f
a
g
r
ap
h
,
th
e
r
esu
lt,
(
,
)
+
|
(
)
−
(
)
|
≥
1
+
∀
,
∈
(
)
,
is
o
b
tain
ed
.
T
h
e
r
ad
io
n
u
m
b
er
s
of
an
d
A
a
r
e
r
ep
r
esen
ted
b
y
(
)
an
d
(
)
wh
ich
ar
e
th
e
m
ax
im
u
m
n
u
m
b
er
ass
ig
n
ed
to
an
y
v
e
r
tex
o
f
an
d
th
e
m
in
i
m
u
m
v
alu
e
o
f
(
)
tak
en
o
v
er
all
la
b
ellin
g
s
k
o
f
,
r
esp
ec
tiv
ely
.
T
h
e
f
o
llo
win
g
Fig
u
r
e
1
d
e
p
icts
th
e
d
ef
in
itio
n
o
f
r
ad
i
o
n
u
m
b
er
.
Fo
r
th
e
last
two
d
ec
ad
es,
s
ev
er
al
au
th
o
r
s
s
tu
d
ied
th
e
r
ad
io
l
ab
ellin
g
p
r
o
b
lem
f
o
r
g
en
e
r
al
g
r
ap
h
s
an
d
ce
r
tain
in
ter
co
n
n
ec
tio
n
n
etwo
r
k
s
.
T
h
e
r
a
d
io
n
u
m
b
e
r
o
f
th
e
to
t
al
p
ath
o
f
g
r
a
p
h
s
wer
e
d
eter
m
i
n
ed
b
y
Vaid
y
a
an
d
B
an
tv
a
[
1
4
]
.
C
ad
a
et
a
l.
[
1
5
]
o
b
tain
ed
th
e
r
ad
io
n
u
m
b
er
o
f
d
is
tan
ce
g
r
ap
h
s
.
T
h
e
s
am
e
p
r
o
b
lem
f
o
r
tr
ee
s
was
s
tu
d
ied
b
y
L
iu
[
1
6
]
.
K
i
m
et
a
l
.
[
1
7
]
p
r
e
s
e
n
te
d
th
e
p
r
o
d
u
ct
o
f
g
r
ap
h
s
n
a
m
e
ly
(
≥
4
)
a
n
d
(
≥
2
)
.
B
h
ar
ati
an
d
Yen
o
k
e
[
1
8
]
d
eter
m
in
ed
b
o
th
u
p
p
er
an
d
lo
we
r
b
o
u
n
d
s
f
o
r
th
e
h
ex
a
g
o
n
al
m
esh
as
(
3
2
−
4
−
1
)
+
3
an
d
3
2
−
3
+
2
+
∑
(
−
−
1
)
−
2
=
0
,
r
esp
ec
tiv
ely
.
B
a
n
t
v
a
[
1
9
]
s
l
i
g
h
t
l
y
i
m
p
r
o
v
e
d
t
h
e
lo
we
r
b
o
u
n
d
th
a
t
w
a
s
e
s
t
a
b
l
i
s
h
e
d
in
[
2
0
]
.
I
n
ad
d
i
t
io
n
,
Y
en
o
k
e
et
a
l.
[
2
1
]
p
r
o
v
e
d
t
h
at
(
(
,
)
)
≤
(
−
2
)
(
4
2
−
9
+
8
)
+
2
(
−
1
)
2
+
(
+
1
)
,
wh
er
e
(
,
)
is
th
e
en
h
an
ce
d
m
esh
,
≥
4
.
T
h
is
p
ap
er
is
d
iv
id
ed
as
f
o
l
lo
ws:
I
n
s
ec
tio
n
2
,
we
d
is
cu
s
s
th
e
m
eth
o
d
o
lo
g
y
o
f
o
u
r
r
esear
ch
wo
r
k
.
I
n
s
ec
tio
n
3
,
o
u
r
m
ai
n
r
es
u
lts
ar
e
o
b
tain
e
d
b
y
s
tu
d
y
in
g
t
h
e
b
o
u
n
d
s
f
o
r
t
h
e
L(
2
,
1
)
-
la
b
ellin
g
n
u
m
b
e
r
a
n
d
r
ad
i
o
n
u
m
b
er
o
f
th
e
g
en
e
r
al
tr
ee
d
en
d
r
im
er
,
.
Ou
r
r
esear
ch
wo
r
k
is
co
n
clu
d
e
d
in
s
ec
tio
n
4
.
Fig
u
r
e
1
.
Dif
f
e
r
en
t r
a
d
io
lab
el
lin
g
s
an
d
r
a
d
io
n
u
m
b
er
o
f
a
g
r
ap
h
A
,
(
)
=
min
{
8
,
6
,
5
,
4
}
=
4
2.
RE
S
E
ARCH
M
E
T
H
O
D
T
h
e
au
th
o
r
s
tu
d
ied
in
[
1
8
]
,
[
2
1
]
th
e
s
am
e
p
r
o
b
lem
f
o
r
th
e
n
et
wo
r
k
s
th
at
co
n
tain
s
th
e
n
u
m
b
e
r
o
f
v
er
tices
in
th
e
n
t
h
d
im
en
s
io
n
as
3
2
−
3
+
1
a
n
d
2
,
r
esp
ec
tiv
ely
.
I
n
ad
d
itio
n
,
th
e
au
th
o
r
s
in
[
1
4
]
,
[
1
5
]
,
[
1
7
]
s
tu
d
ied
th
e
s
am
e
p
r
o
b
lem
f
o
r
th
e
g
r
ap
h
s
with
v
er
tices.
Sin
ce
t
h
e
v
e
r
tices
h
av
e
b
ee
n
in
cr
ea
s
ed
in
t
er
m
s
o
f
t
h
e
h
ig
h
o
r
d
er
f
o
r
a
n
etwo
r
k
,
esp
ec
ially
o
f
o
r
d
er
;
the
r
e
for
e
,
f
in
d
in
g
a
g
o
o
d
s
o
lu
tio
n
is
v
er
y
co
m
p
licated
.
T
h
e
au
t
h
o
r
s
ar
e
tr
y
in
g
to
f
in
d
a
s
o
lu
tio
n
f
o
r
s
u
ch
n
etwo
r
k
s
.
Ho
wev
er
,
a
g
o
o
d
b
o
u
n
d
is
o
b
tain
ed
in
th
i
s
p
ap
er
f
o
r
th
e
tr
ee
d
en
d
r
im
e
r
ch
e
m
ical
n
etwo
r
k
wh
ich
g
r
o
ws
in
t
h
e
o
r
d
er
o
f
g
en
er
atio
n
.
Fu
r
th
e
r
,
m
o
s
t
o
f
t
h
e
n
etwo
r
k
s
wer
e
s
tu
d
ied
s
ep
ar
ately
f
o
r
th
e
L
(
2
,
1
)
lab
ellin
g
o
r
r
ad
io
la
b
ellin
g
.
Du
e
to
th
e
e
x
p
o
n
en
tial g
r
o
wth
o
f
co
m
m
u
n
icatio
n
tech
n
o
lo
g
y
,
we
ar
e
to
d
a
y
in
n
ee
d
to
esti
m
ate
th
e
lo
wer
an
d
u
p
p
e
r
b
o
u
n
d
s
f
o
r
th
e
g
r
a
p
h
s
th
at
ar
e
g
r
o
win
g
in
h
ig
h
er
o
r
d
er
t
o
co
m
p
ete
with
t
h
e
co
n
s
u
m
er
s
’
d
em
an
d
.
B
y
tak
in
g
th
is
in
t
o
o
u
r
ac
co
u
n
t
an
d
a
cc
o
r
d
in
g
to
th
e
b
est
o
f
o
u
r
k
n
o
wled
g
e,
f
o
r
th
e
f
i
r
s
t
tim
e
ev
er
,
we
h
av
e
esti
m
ated
in
th
is
r
esear
ch
wo
r
k
th
e
b
o
u
n
d
s
f
o
r
b
o
th
L
(
2
,
1
)
-
lab
ellin
g
an
d
r
ad
i
o
lab
ellin
g
n
u
m
b
er
s
f
o
r
an
(
n
u
m
b
er
o
f
v
e
r
tices)
ex
p
a
n
d
in
g
ch
e
m
ical
s
tr
u
ctu
r
e
,
k
n
o
wn
as
n
an
ao
s
tar
tr
ee
d
e
n
d
r
im
er
.
T
h
is
r
esear
ch
s
tu
d
y
p
r
o
v
id
es
a
d
etailed
an
aly
s
is
o
f
th
e
g
r
o
wth
o
f
s
u
c
h
g
r
ap
h
i
n
ter
m
s
o
f
d
iam
eter
an
d
v
er
tices
f
o
r
,
>
2
,
an
d
its
b
o
u
n
d
s
h
av
e
b
ee
n
o
b
tain
e
d
s
ep
ar
ately
f
o
r
(
2
,
1
)
-
lab
ellin
g
n
u
m
b
er
an
d
r
ad
i
o
la
b
ellin
g
n
u
m
b
er
.
T
h
e
r
ef
o
r
e
,
a
ll
o
b
tain
ed
r
esu
lts
in
th
is
s
tu
d
y
a
r
e
n
o
v
el
an
d
wo
r
th
y
.
r
n
(
k
)
=
8
0
2
4
6
8
1
4
2
0
6
2
2
r
n
(
k
)
=
6
5
2
0
3
1
3
r
n
(
k
)
=
5
0
3
1
4
2
r
n
(
k
)
=
4
4
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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N
:
1
6
9
3
-
6
9
3
0
TEL
KOM
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KA
T
elec
o
m
m
u
n
C
o
m
p
u
t E
l
C
o
n
tr
o
l
,
Vo
l.
20
,
No
.
1
,
Feb
r
u
ar
y
20
22
:
52
-
60
54
2
.
1
.
Na
no
s
t
a
r
t
re
e
dend
rim
e
r
Nan
o
s
tar
is
a
s
tar
-
l
o
o
k
in
g
ty
p
e
o
f
n
an
o
p
ar
ticle
th
at
c
o
n
tain
s
a
s
p
h
er
ical
co
r
e
with
m
a
n
y
b
r
an
ch
es.
Den
d
r
im
er
s
h
av
e
v
er
y
co
m
p
l
ex
ch
em
ical
s
tr
u
ctu
r
es
an
d
h
y
p
er
-
b
r
a
n
ch
ed
m
a
cr
o
m
o
lecu
les
with
a
s
tar
-
s
h
ap
ed
ar
ch
itectu
r
e.
I
n
ad
d
itio
n
,
d
e
n
d
r
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s
ar
e
class
if
ied
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wh
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e
p
r
esen
ts
th
e
r
ep
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b
r
a
n
ch
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g
cy
cles
n
u
m
b
er
t
h
at
ar
e
p
er
f
o
r
m
ed
d
u
r
in
g
its
s
y
n
th
esis
.
T
h
e
s
tr
u
ctu
r
e
o
f
th
ese
m
ater
ials
h
a
s
a
h
u
g
e
im
p
ac
t
o
n
th
e
p
h
y
s
ical
an
d
ch
em
ical
p
r
o
p
er
ties
o
f
d
en
d
r
im
er
s
d
u
e
to
th
e
u
n
iq
u
e
n
ess
o
f
d
en
d
r
im
er
s
’
b
eh
av
io
r
wh
ic
h
m
a
k
es
th
em
v
er
y
s
u
itab
le
f
o
r
v
ar
i
o
u
s
b
io
m
ed
ical
an
d
in
d
u
s
tr
ial
ap
p
licatio
n
s
[
2
2
]
-
[
2
4
]
.
Yan
g
an
d
Xia
[
2
4
]
d
ef
in
ed
a
tr
ee
d
en
d
r
im
er
g
r
ap
h
,
d
en
o
ted
b
y
,
,
as
f
o
llo
ws:
T
h
e
ce
n
te
r
v
er
tex
o
f
th
e
g
r
ap
h
,
is
r
ep
r
esen
ted
b
y
1
0
wh
ich
is
a
−
r
eg
u
lar
g
r
a
p
h
ex
ce
p
t th
e
p
en
d
an
t v
er
tices.
I
n
ad
d
it
io
n
,
th
e
d
is
tan
ce
f
r
o
m
th
e
ce
n
t
er
v
er
tex
1
0
to
ev
er
y
p
e
n
d
an
t
v
er
tex
is
ex
ac
tly
.
Mo
r
eo
v
er
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ig
n
if
ies
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er
e
th
e
ℎ
g
en
er
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o
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th
e
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r
ee
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en
d
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im
er
.
T
h
e
d
iam
eter
an
d
r
ad
iu
s
o
f
a
tr
ee
d
en
d
r
im
er
g
r
ap
h
ar
e
2
an
d
,
r
esp
ec
tiv
ely
.
I
n
th
is
r
esear
ch
wo
r
k
,
we
h
av
e
n
am
ed
th
e
g
en
er
atio
n
v
er
tice
s
o
f
th
e
tr
ee
d
en
d
r
im
er
,
as f
o
llo
ws
:
First,
we
n
am
e
t
h
e
v
er
tices
i
n
th
e
f
ir
s
t
g
e
n
er
atio
n
wh
ich
ar
e
ad
jace
n
t
to
th
e
ce
n
ter
v
er
tex
1
0
a
s
1
1
,
2
1
…
1
in
th
e
clo
ck
wis
e
s
en
s
e
.
Nex
t,
we
n
am
e
th
e
(
−
1
)
v
er
tices
in
th
e
s
ec
o
n
d
g
e
n
er
atio
n
as
1
2
,
2
2
…
(
−
1
)
2
in
th
e
s
am
e
o
r
d
er
as
we
d
id
th
e
p
r
ev
io
u
s
s
tep
.
Similar
ly
,
we
n
am
e
th
e
v
e
r
tices
o
f
3
,
4
ℎ
…
(
−
1
)
ℎ
g
en
e
r
atio
n
s
.
F
i
n
a
l
l
y
,
t
h
e
(
−
1
)
−
1
v
e
r
t
i
c
e
s
i
n
t
h
e
ℎ
g
e
n
e
r
a
t
i
o
n
a
r
e
n
a
m
e
d
as
1
,
2
…
(
−
1
)
−
1
a
s
s
h
o
w
n
i
n
F
i
g
u
r
e
2
.
Fig
u
r
e
2
.
Ver
tices’
Nam
in
g
in
3
,
3
3.
RE
SU
L
T
S
A
ND
AN
AL
Y
SI
S
T
h
e
b
o
u
n
d
s
f
o
r
t
h
e
L(2
,
1
)
-
lab
ellin
g
n
u
m
b
er
a
n
d
r
a
d
io
n
u
m
b
er
o
f
t
h
e
g
e
n
er
al
tr
ee
d
en
d
r
im
er
,
ar
e
d
eter
m
in
ed
in
th
is
s
ec
tio
n
.
Pro
p
o
s
itio
n
3
.
1
:
Fo
r
an
y
co
n
n
ec
ted
s
im
p
le
g
r
ap
h
o
f
d
iam
et
er
2
,
2
,
1
(
)
+
1
=
(
)
.
Pro
o
f
:
T
h
e
p
r
o
o
f
is
d
ir
ec
tly
d
e
r
iv
ed
f
r
o
m
th
e
d
ef
in
itio
n
s
o
f
L
(
2
,
1
)
-
lab
ellin
g
an
d
r
a
d
io
lab
ell
in
g
.
a)
T
h
eo
r
em
3
.
1
:
Fo
r
>
2
,
2
,
1
(
1
,
)
+
1
=
(
1
,
)
=
+
2
.
Pro
o
f
:
I
t
is
k
n
o
wn
th
at
th
e
tr
ee
d
en
d
r
im
er
1
,
is
an
o
r
d
in
ar
y
s
tar
g
r
ap
h
+
1
.
R
ajan
et
a
l.
[
2
5
]
s
h
o
wed
th
at
th
e
r
ad
io
n
u
m
b
er
o
f
a
s
tar
g
r
ap
h
is
(
+
1
)
=
+
2
,
>
2
.
Sin
ce
th
e
d
iam
eter
o
f
1
,
is
2
,
f
r
o
m
Pr
o
p
o
s
itio
n
3
.
1
,
we
o
b
tain
2
,
1
(
1
,
)
+
1
=
(
1
,
)
=
+
2
,
>
2
.
b)
T
h
eo
r
em
3
.
2
:
Fo
r
=
2
a
n
d
>
2
,
th
e
L
(
2
,
1
)
l
ab
ellin
g
n
u
m
b
er
o
f
,
s
atis
f
ies,
2
,
1
(
2
,
)
≤
2
.
Pro
o
f
:
B
y
d
ef
in
i
n
g
a
m
ap
p
in
g
:
(
2
,
)
→
,
we
h
av
e
th
e
f
o
llo
win
g
:
(
1
0
)
=
1
(
(
−
1
)
(
−
1
)
+
2
)
=
+
1
,
=
1
,
2
…
−
1
,
=
1
,
2
…
(
1
)
=
+
1
+
,
=
1
,
2
…
.
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KOM
NI
KA
T
elec
o
m
m
u
n
C
o
m
p
u
t E
l Co
n
tr
o
l
Th
e
b
o
u
n
d
s
fo
r
th
e
d
is
ta
n
ce
t
w
o
la
b
ellin
g
a
n
d
r
a
d
io
la
b
elli
n
g
o
f n
a
n
o
s
ta
r
tr
ee
d
en
d
r
ime
(
K
in
s
Yen
o
ke
)
55
Nex
t,
we
claim
th
at
is
a
v
alid
r
ad
io
2
-
ch
r
o
m
atic
lab
ellin
g
.
L
et
an
d
b
e
an
y
two
v
er
tices i
n
2
,
.
1)
C
ase
1
:
As
s
u
m
e
an
d
ar
e
an
y
t
wo
s
ec
o
n
d
g
e
n
er
atio
n
v
er
tices.
If
=
(
−
1
)
(
−
1
)
+
2
an
d
=
(
−
1
)
(
−
1
)
+
2
,
1
≤
≠
≤
−
1
,
th
en
we
h
av
e
(
,
)
=
2
an
d
|
(
)
−
(
)
|
=
1
.
T
h
er
ef
o
r
e,
we
h
av
e:
(
,
)
+
|
(
)
−
(
)
|
≥
3
.
I
n
ad
d
itio
n
,
if
=
(
−
1
)
(
−
1
)
+
2
an
d
=
(
−
1
)
(
−
1
)
+
2
,
1
≤
≠
≤
;
h
en
ce
,
th
e
d
is
tan
ce
b
etwe
en
th
em
is
4
.
T
h
er
ef
o
r
e,
th
e
c
o
n
d
itio
n
is
tr
iv
ially
s
atis
f
ied
.
2)
C
ase
2
:
Su
p
p
o
s
e
c
an
d
s
ar
e
t
h
e
f
ir
s
t
-
g
e
n
er
atio
n
v
er
tices,
th
en
(
,
)
=
2
an
d
(
)
=
+
1
+
,
(
)
=
+
1
+
,
1
≤
≠
≤
.
T
h
er
ef
o
r
e,
we
o
b
tain
(
,
)
+
|
(
)
−
(
)
|
≥
2
+
|
−
|
≥
3
.
3)
C
ase
3
:
Su
p
p
o
s
e
c
is
a
s
ec
o
n
d
-
g
en
er
atio
n
v
er
tex
an
d
s
is
a
f
ir
s
t
-
g
en
er
atio
n
v
e
r
tex
,
th
en
we
h
av
e:
(
,
)
≥
1
an
d
(
)
=
+
1
,
(
)
=
+
1
+
,
1
≤
≤
−
1
,
1
≤
≤
.
T
h
er
ef
o
r
e
,
(
,
)
+
|
(
)
−
(
)
|
≥
1
+
|
|
≥
3
.
4)
C
ase
4
:
If
c
is
th
e
ce
n
ter
v
er
te
x
an
d
=
(
−
1
)
(
−
1
)
+
2
,
th
en
(
,
)
=
2
an
d
|
(
)
−
(
)
|
≥
1
.
T
h
er
ef
o
r
e,
we
o
b
tain
:
(
,
)
+
|
(
)
−
(
)
|
≥
3
.
5)
C
ase
5
:
If
=
1
0
an
d
=
1
,
th
en
(
)
=
1
,
(
)
≥
+
2
.
T
h
e
r
ef
o
r
e,
(
,
)
+
|
(
)
−
(
)
|
≥
+
3
>
3
.
Hen
ce
,
th
e
(
2
,
1
)
lab
ellin
g
c
o
n
d
itio
n
is
s
atis
f
ied
,
an
d
th
e
v
e
r
tex
1
a
ttain
s
th
e
m
ax
im
u
m
v
alu
e
2
+
1
.
Sin
ce
we
s
tar
t la
b
ellin
g
f
r
o
m
1
,
we
g
et
2
,
1
(
2
,
)
+
1
≤
2
+
1
,
wh
ich
im
p
lies
th
at
2
,
1
(
2
,
)
≤
2
.
c)
T
h
eo
r
em
3
.
3
:
L
et
b
e
a
tr
ee
d
e
n
d
r
im
er
,
,
whe
r
e
,
>
2
,
th
en
th
e
L
(
2
,
1
)
la
b
ellin
g
n
u
m
b
er
o
f
A
s
atis
f
ies
2
,
1
(
)
≤
3
+
1
.
Pro
o
f
: D
ef
in
e
a
m
ap
p
in
g
:
(
,
)
→
as f
o
llo
ws:
(
1
0
)
=
1
.
(
1
)
=
+
2
,
=
1
,
2
…
.
(
(
−
1
)
(
−
1
)
+
3
−
1
)
=
+
3
+
,
=
1
,
2
…
−
1
,
=
1
,
2
…
(
−
1
)
3
(
−
1
)
,
=
1
,
2
…
[
3
]
.
(
(
−
1
)
(
−
1
)
+
3
)
=
2
+
3
+
,
=
1
,
2
…
−
1
,
=
1
,
2
…
(
−
1
)
3
−
2
,
=
1
,
2
…
⌊
3
⌋
.
(
(
−
1
)
(
−
1
)
+
3
+
1
)
=
+
2
,
=
1
,
2
…
−
1
,
=
1
,
2
…
(
−
1
)
3
,
=
1
,
2
…
⌈
3
⌉
−
1
a
s
s
h
o
w
n
i
n
F
i
g
u
r
e
3
.
Nex
t,
we
v
er
if
y
th
e
r
ad
io
2
-
ch
r
o
m
atic
lab
ellin
g
co
n
d
itio
n
(
,
)
+
|
(
)
−
(
)
|
≥
3
∀
,
∈
(
,
)
.
B
y
lettin
g
,
∈
(
,
)
,
we
h
av
e
th
e
f
o
llo
win
g
:
1)
C
ase
1
:
Su
p
p
o
s
e
an
d
ar
e
an
y
two
v
er
tices in
(
3
−
1
)
ℎ
g
en
er
atio
n
,
w
h
er
e
=
1
,
2
…
[
3
]
.
−
C
ase
1
.
1
:
If
=
(
−
1
)
(
−
1
)
+
3
−
1
an
d
=
(
−
1
)
(
−
1
)
+
3
−
1
,
1
≤
≠
≤
[
3
]
,
th
en
we
h
av
e:
(
,
)
≥
3
a
n
d
|
(
)
−
(
)
|
≥
0
.
The
r
e
f
or
e
,
we
ob
ta
in
:
(
,
)
+
|
(
)
−
(
)
|
≥
3
.
−
C
ase
1
.
2
:
I
f
=
(
−
1
)
(
−
1
)
+
3
−
1
an
d
=
(
−
1
)
(
−
1
)
+
3
−
1
,
1
≤
≠
≤
(
−
1
)
3
(
−
1
)
,
th
e
n
w
e
h
av
e:
(
,
)
≥
4
a
n
d
|
(
)
−
(
)
|
≥
0
.
The
r
e
f
or
e
,
we
ob
ta
in
(
,
)
+
|
(
)
−
(
)
|
>
3
−
C
ase
1
.
3
:
If
=
(
−
1
)
(
−
1
)
+
3
−
1
an
d
=
(
−
1
)
(
−
1
)
+
3
−
1
,
1
≤
≠
≤
−
1
,
th
en
we
h
av
e:
−
(
,
)
=
2
a
n
d
|
(
)
−
(
)
|
=
|
(
+
3
+
)
−
(
+
3
+
)
|
=
|
−
|
≥
1
,
s
in
c
e
≠
.
Hen
ce
,
we
h
av
e
(
,
)
+
|
(
)
−
(
)
|
≥
2
+
1
=
3
.
2)
C
ase
2
:
L
et
u
s
ass
u
m
e
th
at
an
d
lie
in
th
e
(
3
)
ℎ
g
en
er
atio
n
,
wh
er
e
v
ar
ies f
r
o
m
1
to
⌊
3
⌋
.
−
C
ase
2
.
1
:
If
a
n
d
a
r
e
of
the
for
m
(
−
1
)
(
−
1
)
+
3
an
d
(
−
1
)
(
−
1
)
+
3
,
1
≤
≠
≤
⌊
3
⌋
,
th
en
th
e
d
is
tan
ce
b
etwe
en
th
em
is
at
le
ast 3
,
wh
ich
d
ir
ec
tly
v
er
if
ies th
e
r
ad
io
2
-
ch
r
o
m
atic
lab
ellin
g
co
n
d
itio
n
.
−
C
ase
2
.
2
:
If
=
(
−
1
)
(
−
1
)
+
3
an
d
=
(
−
1
)
(
−
1
)
+
3
,
1
≤
≠
≤
(
−
1
)
3
−
2
,
th
en
(
,
)
≥
4
.
The
r
e
fo
r
e
,
(
,
)
+
|
(
)
−
(
)
|
>
3
.
−
C
ase
2
.
3
:
I
f
=
(
−
1
)
(
−
1
)
+
3
an
d
=
(
−
1
)
(
−
1
)
+
3
,
1
≤
≠
≤
−
1
,
th
en
(
,
)
=
2
a
n
d
|
(
)
−
(
)
|
=
|
(
2
+
3
+
)
−
(
2
+
3
+
)
|
.
Hen
ce
,
(
,
)
+
|
(
)
−
(
)
|
≥
2
+
|
−
|
≥
3
,
s
in
c
e
≠
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
1
6
9
3
-
6
9
3
0
TEL
KOM
NI
KA
T
elec
o
m
m
u
n
C
o
m
p
u
t E
l
C
o
n
tr
o
l
,
Vo
l.
20
,
No
.
1
,
Feb
r
u
ar
y
20
22
:
52
-
60
56
Fig
u
r
e
3
.
R
ad
io
2
-
ch
r
o
m
atic
l
ab
ellin
g
o
f
,
with
=
=
4
wh
ich
attain
s
th
e
u
p
p
er
b
o
u
n
d
3)
C
ase
3
:
As
s
u
m
e
th
at
an
d
ar
e
(
3
+
1
)
ℎ
g
en
er
atio
n
v
er
tices,
wh
er
e
=
1
,
2
…
[
3
]
.
−
C
ase
3
.
1
:
If
=
(
−
1
)
(
−
1
)
+
3
+
1
an
d
=
(
−
1
)
(
−
1
)
+
3
+
1
,
1
≤
≠
≤
⌈
3
⌉
−
1
,
th
en
|
(
)
−
(
)
|
≥
0
an
d
th
e
d
is
tan
ce
b
etwe
en
th
em
is
g
r
ea
ter
th
an
2
.
The
r
e
for
e
,
(
,
)
+
|
(
)
−
(
)
|
≥
3
.
−
C
ase
3
.
2
:
If
=
(
−
1
)
(
−
1
)
+
3
+
1
an
d
=
(
−
1
)
(
−
1
)
+
3
+
1
,
1
≤
≠
≤
(
−
1
)
3
,
th
en
we
h
av
e:
(
,
)
≥
4
,
whic
h
tr
ivia
l
l
y
ve
r
ifi
e
s
the
l
a
b
e
l
l
in
g
c
on
diti
o
n
.
−
C
ase
3
.
3
:
I
f
=
(
−
1
)
(
−
1
)
+
3
−
1
an
d
=
(
−
1
)
(
−
1
)
+
3
−
1
,
th
en
we
h
a
v
e
(
)
=
+
2
an
d
(
)
=
+
2
,
1
≤
≠
≤
−
1
.
I
n
a
d
d
itio
n
,
th
e
d
is
tan
ce
b
e
twee
n
th
em
is
ex
ac
tly
2
.
Hen
ce
,
we
o
b
tain
th
e
f
o
llo
win
g
:
(
,
)
+
|
(
)
−
(
)
|
≥
3
.
4)
C
ase
4
:
Su
p
p
o
s
e
an
d
ar
e
fir
s
t
ge
n
e
r
a
tio
n
ve
r
tic
e
s
the
n
,
(
)
=
+
2
an
d
(
)
=
+
2
,
1
≤
≠
≤
.
I
n
ad
d
itio
n
,
in
th
is
ca
s
e
(
,
)
=
2
.
T
h
e
r
ef
o
r
e,
(
,
)
+
|
(
)
−
(
)
|
≥
2
+
|
−
|
≥
3
s
in
ce
≠
.
5)
C
as
e
5
:
S
u
p
p
o
s
e
=
1
a
n
d
=
(
−
1
)
(
−
1
)
+
3
−
1
,
t
h
e
n
(
)
=
+
2
,
1
≤
≤
an
d
(
)
=
+
3
+
,
1
≤
≤
−
1
.
I
n
a
d
d
i
t
i
o
n
,
(
,
)
≥
1
.
H
e
n
c
e
,
(
,
)
+
|
(
)
−
(
)
|
≥
1
+
|
(
+
3
+
1
)
−
(
+
2
)
|
=
3
.
6)
C
ase
6
:
Su
p
p
o
s
e
=
1
,
1
≤
≤
an
d
=
(
−
1
)
(
−
1
)
+
3
,
1
≤
≤
−
1
,
1
≤
≤
(
−
1
)
3
(
−
1
)
,
1
≤
≤
[
3
]
,
th
en
(
)
=
2
+
,
an
d
(
)
=
2
+
3
+
.
Als
o
,
th
e
d
is
t
an
ce
b
etwe
en
th
em
is
at
lea
s
t
2
.
T
h
er
ef
o
r
e,
(
,
)
+
|
(
)
−
(
)
|
≥
2
+
|
(
2
+
3
+
1
)
−
(
+
2
)
|
=
+
4
>
3
.
7)
C
ase
7
:
Su
p
p
o
s
e
=
1
,
1
≤
≤
an
d
=
(
−
1
)
(
−
1
)
+
3
+
1
,
1
≤
≤
−
1
,
1
≤
≤
(
−
1
)
3
,
1
≤
≤
⌈
3
⌉
−
1
,
th
en
|
(
)
−
(
)
|
≥
0
.
Ho
wev
er
,
th
e
d
is
tan
ce
b
etwe
en
th
em
is
at
least 3
.
He
n
ce
,
we
o
b
tain
:
(
,
)
+
|
(
)
−
(
)
|
≥
3
.
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KOM
NI
KA
T
elec
o
m
m
u
n
C
o
m
p
u
t E
l Co
n
tr
o
l
Th
e
b
o
u
n
d
s
fo
r
th
e
d
is
ta
n
ce
t
w
o
la
b
ellin
g
a
n
d
r
a
d
io
la
b
elli
n
g
o
f n
a
n
o
s
ta
r
tr
ee
d
en
d
r
ime
(
K
in
s
Yen
o
ke
)
57
8)
C
ase
8
:
Su
p
p
o
s
e
=
(
−
1
)
(
−
1
)
+
3
−
1
an
d
=
(
−
1
)
(
−
1
)
+
3
,
1
≤
≤
−
1
,
th
en
w
e
h
av
e
(
)
=
+
3
+
,
(
)
=
2
+
3
+
an
d
(
,
)
≥
1
.
T
h
er
ef
o
r
e,
we
o
b
tain
:
(
,
)
+
|
(
)
−
(
)
|
≥
1
+
|
(
2
+
3
+
1
)
−
(
2
+
2
)
|
=
3
.
9)
C
ase
9
:
Su
p
p
o
s
e
=
(
−
1
)
(
−
1
)
+
3
−
1
an
d
=
(
−
1
)
(
−
1
)
+
3
+
1
,
1
≤
≤
−
1
,
th
en
(
)
=
+
3
+
,
(
)
=
+
2
an
d
(
,
)
≥
2
.
T
h
er
ef
o
r
e,
we
h
a
v
e:
(
,
)
+
|
(
)
−
(
)
|
≥
2
+
|
(
+
4
)
−
(
+
1
)
)
|
>
3
.
10)
C
ase
1
0
:
If
=
(
−
1
)
(
−
1
)
+
3
an
d
=
(
−
1
)
(
−
1
)
+
3
+
1
,
1
≤
≤
−
1
,
th
en
we
g
et:
(
)
=
2
+
3
+
,
(
)
=
+
2
an
d
(
,
)
≥
2
.
T
h
er
ef
o
r
e,
(
,
)
+
|
(
)
−
(
)
|
≥
2
+
|
(
2
+
4
)
−
(
+
1
)
)
|
=
+
5
>
3
.
11)
C
a
s
e
11
:
If
is
th
e
ce
n
ter
v
er
tex
1
0
,
an
d
is
an
y
o
th
er
v
er
tex
in
th
e
g
r
ap
h
.
T
h
e
n
,
(
,
)
≥
1
an
d
|
(
)
−
(
)
|
≥
2
.
T
h
er
ef
o
r
e,
we
o
b
tain
:
(
,
)
+
|
(
)
−
(
)
|
≥
3
.
Hen
ce
,
(
,
)
+
|
(
)
−
(
)
|
≥
3
f
o
r
ev
er
y
p
air
o
f
v
e
r
tices
c
an
d
s
in
(
,
)
.
I
n
ad
d
itio
n
,
t
h
e
v
er
tices
(
−
1
)
3
,
=
1
,
2
…
(
−
1
)
3
−
2
,
=
1
,
2
…
⌊
3
⌋
attain
s
th
e
m
ax
im
u
m
v
al
u
e
3
+
2
,
whic
h
imp
l
ie
s
tha
t
2
,
1
(
)
+
1
≤
3
+
2
.
T
h
u
s
,
2
,
1
(
)
≤
3
+
1
.
R
em
ar
k
1
:
T
h
e
b
r
an
c
h
es
o
f
th
e
tr
ee
d
en
d
r
im
er
w
h
ich
ar
e
co
n
n
ec
ted
to
th
e
ce
n
ter
v
e
r
tex
1
0
b
y
a
s
in
g
le
ed
g
e
ar
e
ca
lled
th
e
m
ain
b
r
an
ch
es
o
f
th
e
tr
ee
d
en
d
r
im
er
g
r
ap
h
.
W
e
d
en
o
te
th
e
m
ain
b
r
an
ch
es
in
,
as
,
=
1
,
2
…
.
L
em
m
a
3
.
1
:
L
et
,
b
e
a
tr
e
e
d
e
n
d
r
im
er
g
r
ap
h
o
f
g
en
er
atio
n
s
with
ea
ch
v
er
te
x
o
f
d
eg
r
ee
ex
p
ec
ts
th
e
p
en
d
an
t
v
er
tices,
th
en
th
e
n
u
m
b
er
o
f
v
er
tices in
ea
ch
m
ain
b
r
an
ch
(
1
≤
≤
)
is
(
−
1
)
−
1
−
2
.
Pro
o
f
:
Fro
m
th
e
co
n
s
tr
u
ctio
n
o
f
th
e
tr
ee
d
en
d
r
im
e
r
,
th
e
f
ir
s
t g
e
n
er
atio
n
co
n
tain
s
−
1
v
er
tices.
Sin
ce
th
e
r
o
o
t
v
er
tex
o
f
a
m
ain
b
r
an
ch
is
a
v
er
tex
o
f
f
ir
s
t
g
en
e
r
atio
n
,
th
e
r
e
is
o
n
l
y
a
s
in
g
le
v
er
tex
in
t
h
e
f
ir
s
t
g
en
er
atio
n
.
T
h
er
ef
o
r
e,
th
e
n
u
m
b
er
o
f
v
e
r
tices
in
a
s
ec
o
n
d
g
en
er
atio
n
is
−
1
.
I
n
g
e
n
er
al,
th
e
n
u
m
b
e
r
o
f
v
er
ti
ce
s
in
th
e
ℎ
g
en
er
atio
n
is
(
−
1
)
−
1
.
Hen
ce
,
th
e
to
tal
n
u
m
b
e
r
o
f
v
er
tices in
a
m
ain
b
r
an
ch
is
ca
lcu
lated
as f
o
llo
ws:
1
+
(
−
1
)
+
(
−
1
)
2
+
⋯
(
−
1
)
−
1
=
(
−
1
)
−
`1
−
1
−
1
=
(
−
1
)
−
1
−
2
.
d)
T
h
eo
r
em
3
.
4
:
L
et
,
(
,
>
2
)
b
e
a
tr
ee
d
en
d
r
im
er
g
r
a
p
h
o
f
d
iam
eter
2
.
T
h
en
,
a
n
u
p
p
e
r
b
o
u
n
d
f
o
r
t
h
e
r
ad
io
n
u
m
b
er
o
f
,
is
g
iv
en
b
y
(
,
)
≤
+
(
2
−
1
)
+
1
+
∑
(
2
−
1
)
(
(
−
1
)
−
−
1
)
)
(
−
−
1
=
1
2
)
+
(
2
−
1
)
(
)
−
−
1
−
1
,
whe
n
e
ve
r
≥
2
−
3
.
Pro
o
f
:
Def
in
e
a
m
ap
p
in
g
ℎ
f
r
o
m
th
e
v
er
tex
s
et
o
f
,
to
th
e
n
atu
r
al
n
u
m
b
e
r
s
as f
o
llo
ws:
First,
we
lab
el
th
e
ce
n
ter
v
er
tex
1
0
as
ℎ
(
1
0
)
=
1
.
Sin
ce
th
e
p
en
d
an
t
v
er
tice
s
ar
e
at
a
d
i
s
tan
ce
f
r
o
m
th
e
ce
n
ter
v
er
tex
,
we
lab
el
th
e
ℎ
g
e
n
er
ati
o
n
p
en
d
a
n
t
v
e
r
tices
in
,
=
1
,
2
…
as
(
(
−
1
)
−
1
(
−
1
)
+
+
(
−
1
)
(
−
1
)
)
=
(
(
−
1
)
−
2
)
(
−
1
)
+
−
1
,
=
1
,
2
…
(
−
1
)
−
2
,
=
1
,
2
…
−
1
.
Nex
t,
we
lab
el
th
e
(
−
1
)
ℎ
g
en
er
atio
n
v
er
tices in
,
=
1
,
2
…
as
(
(
−
1
)
−
2
(
−
1
)
+
+
(
−
1
)
(
−
1
)
−
1
)
=
(
(
−
1
)
−
2
)
(
−
2
)
+
(
(
−
1
)
−
2
)
−
1
+
(
3
(
(
−
1
)
−
3
)
)
(
−
1
)
+
3
−
1
,
=
1
,
2
…
(
−
1
)
−
3
,
=
1
,
2
…
−
1
.
I
n
g
en
e
r
al,
(
−
)
ℎ
(
1
≤
≤
−
1
)
g
en
er
atio
n
v
er
tices
in
,
=
1
,
2
…
ar
e
lab
e
lled
as
(
(
−
1
)
−
(
−
1
)
+
+
(
−
1
)
(
−
1
)
−
)
=
(
(
−
1
)
−
+
1
+
−
1
+
(
−
1
)
(
(
−
1
)
−
−
3
−
1
)
−
+
1
)
+
(
2
−
1
)
(
(
−
1
)
−
−
2
)
)
(
−
1
)
+
(
2
−
1
)
−
1
,
=
1
,
2
…
(
−
1
)
−
−
2
,
=
1
,
2
…
−
1
.
Fin
ally
,
let
u
s
lab
el
th
e
f
ir
s
t
-
g
e
n
er
atio
n
v
e
r
tices o
f
,
as
(
1
)
=
(
(
−
1
)
2
)
+
(
2
−
1
)
−
1
,
=
1
,
2
…
as sh
o
wn
in
Fig
u
r
e
4
.
Giv
en
th
e
d
iam
eter
o
f
t
h
e
g
r
a
p
h
is
2
,
to
p
r
o
v
e
is
a
v
alid
r
a
d
io
la
b
ellin
g
,
we
m
u
s
t
v
e
r
if
y
th
e
co
n
d
itio
n
(
,
)
+
|
(
)
−
(
)
|
≥
2
+
1
f
o
r
ev
er
y
p
air
o
f
v
er
tices
an
d
in
,
.
1)
C
ase
1
:
As
s
u
m
e
an
d
ar
e
an
y
two
v
er
tices in
th
e
s
am
e
(
1
≤
≤
)
,
th
en
we
h
av
e:
(
,
)
=
2
,
=
1
,
2
…
−
1
.
The
r
e
for
e
,
f
o
r
s
u
c
h
ca
s
es,
|
(
)
−
(
)
|
≥
2
−
2
+
1
s
in
c
e
≥
2
−
3
.
Hen
ce
,
we
o
b
tain
:
(
,
)
+
|
(
)
−
(
)
|
≥
|
2
−
(
2
−
2
+
1
)
|
=
2
+
1
.
2)
C
ase
2
:
As
s
u
m
e
is
a
v
er
tex
o
f
an
d
is
a
v
er
tex
o
f
,
1
≤
≠
≤
,
th
en
th
e
d
is
tan
ce
b
etwe
en
th
em
is
ex
ac
tly
2
,
an
d
|
(
)
−
(
)
|
≥
.
T
h
er
ef
o
r
e,
we
h
av
e:
(
,
)
+
|
(
)
−
(
)
|
≥
2
+
1
.
3)
C
ase
3
:
As
s
u
m
e
is
th
e
ce
n
ter
v
er
tex
an
d
is
an
y
o
th
er
v
er
tex
in
,
.
−
C
ase
3
.
1
:
If
is
a
p
en
d
a
n
t
v
er
t
ex
,
th
en
(
,
)
=
an
d
(
)
=
1
,
(
)
≥
+
1
.
T
h
er
e
f
o
r
e,
i
n
th
i
s
s
u
b
ca
s
e,
(
,
)
+
|
(
)
−
(
)
|
≥
2
+
1
.
−
C
ase
3
.
2
:
If
is
n
o
t
a
p
e
n
d
an
t
v
e
r
tex
,
th
en
(
,
)
≥
1
an
d
(
)
=
1
,
(
)
≥
(
−
1
)
2
+
+
1
.
Sin
ce
≥
2
−
3
,
we
o
b
tain
:
(
,
)
+
|
(
)
−
(
)
|
≥
2
+
1
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
1
6
9
3
-
6
9
3
0
TEL
KOM
NI
KA
T
elec
o
m
m
u
n
C
o
m
p
u
t E
l
C
o
n
tr
o
l
,
Vo
l.
20
,
No
.
1
,
Feb
r
u
ar
y
20
22
:
52
-
60
58
T
h
u
s
,
is
a
v
alid
r
ad
io
lab
ellin
g
an
d
t
h
e
v
er
tex
(
1
)
attain
s
th
e
m
ax
im
u
m
v
al
u
e
+
(
2
−
1
)
+
1
+
∑
(
2
−
1
)
(
(
−
1
)
−
−
1
)
)
(
−
2
)
+
(
2
−
1
)
(
)
−
−
1
−
1
−
1
=
1
.
Hen
ce
,
th
e
r
ad
i
o
n
u
m
b
er
o
f
,
(
,
>
2
)
s
atis
f
ies
(
,
)
≤
+
(
2
−
1
)
+
1
+
∑
(
2
−
−
1
=
1
1
)
(
(
−
1
)
−
−
1
)
)
(
−
2
)
+
(
2
−
1
)
(
)
−
−
1
−
1
,
whe
n
e
ve
r
≥
2
−
3
.
Hen
ce
,
th
e
th
eo
r
em
is
p
r
o
v
en
.
Nex
t,
we
d
eter
m
in
e
th
e
lo
wer
b
o
u
n
d
f
o
r
th
e
r
a
d
io
n
u
m
b
e
r
o
f
,
(
,
>
2
)
b
y
u
s
in
g
th
e
f
o
llo
win
g
th
eo
r
em
wh
ich
was p
r
o
v
e
n
b
y
B
h
ar
ati
an
d
Yen
o
k
e
[
1
8
]
:
T
h
eo
r
em
3
.
5
(
As
T
h
eo
r
em
2
in
[
1
8
]
):
L
et
A
b
e
a
s
im
p
le
co
n
n
ec
ted
g
r
ap
h
o
f
o
r
d
e
r
m
.
L
et
0
,
1
…
b
e
th
e
n
u
m
b
er
o
f
v
e
r
tices
th
at
h
av
e
e
cc
en
tr
icities
0
,
1
…
,
wh
er
e
(
)
=
=
0
>
1
>
⋯
>
=
(
)
.
T
h
en
,
we
o
b
tain
th
e
f
o
llo
win
g
:
(
)
≥
{
−
2
(
−
)
+
∑
2
(
−
)
,
>
1
=
1
−
(
−
)
−
(
−
−
1
)
+
∑
2
(
−
)
,
=
1
=
1
.
Fig
u
r
e
4
.
A
r
a
d
io
lab
ellin
g
o
f
3
,
5
wh
ich
attain
s
th
e
u
p
p
e
r
b
o
u
n
d
L
em
m
a
3
.
2
.
Fo
r
th
e
tr
ee
d
en
d
r
i
m
er
g
r
a
p
h
,
,
th
e
ec
ce
n
tr
icities
0
,
1
…
ar
e
g
iv
en
b
y
:
−
1
=
2
−
−
1
,
=
1
,
2
…
+
1
.
Pro
o
f
:
I
t
is
o
b
v
io
u
s
th
at
th
e
d
iam
eter
o
f
th
e
g
r
ap
h
is
2
,
an
d
th
er
e
also
ex
is
ts
at
least
o
n
e
p
ath
1
,
1
−
1
…
1
2
,
1
1
,
1
0
,
2
2
…
(
−
1
)
−
1
−
2
1
wh
ich
p
ass
es
th
r
o
u
g
h
th
e
ce
n
ter
o
f
th
e
g
r
ap
h
.
Hen
ce
,
t
h
e
co
n
s
ec
u
tiv
e
ec
ce
n
tr
icate
s
ar
e
d
if
f
er
en
t
b
y
ex
ac
tly
1
.
T
h
er
ef
o
r
e,
th
e
ec
ce
n
tr
icities
o
f
,
ar
e
0
,
1
…
.
Tha
t
is
,
−
1
=
2
−
−
1
,
=
1
,
2
…
+
1
.
5
6
7
8
9
1
0
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
7
1
9
2
0
2
1
2
2
2
3
2
4
2
5
2
6
2
7
2
8
2
9
3
0
3
1
3
2
3
3
3
4
3
5
3
6
3
7
3
7
3
9
4
0
4
1
4
2
4
3
4
4
4
5
4
6
4
7
4
8
4
9
5
0
5
1
5
2
8
1
5
3
5
4
5
5
5
6
5
7
5
7
5
9
6
0
6
1
6
2
6
3
6
4
6
5
6
6
6
7
6
8
6
9
7
0
7
1
7
2
7
5
8
0
7
6
7
7
7
7
8
3
7
9
8
4
8
6
8
9
9
2
9
5
9
8
1
0
1
1
0
4
1
0
7
1
1
0
1
1
3
1
1
6
1
1
9
1
2
2
1
2
5
1
2
8
1
3
1
1
3
4
1
3
7
1
4
0
1
4
3
1
4
7
1
5
2
1
5
7
1
6
2
1
6
7
1
8
2
8
4
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KOM
NI
KA
T
elec
o
m
m
u
n
C
o
m
p
u
t E
l Co
n
tr
o
l
Th
e
b
o
u
n
d
s
fo
r
th
e
d
is
ta
n
ce
t
w
o
la
b
ellin
g
a
n
d
r
a
d
io
la
b
elli
n
g
o
f n
a
n
o
s
ta
r
tr
ee
d
en
d
r
ime
(
K
in
s
Yen
o
ke
)
59
L
em
m
a
3
.
3
.
Fo
r
th
e
tr
ee
d
en
d
r
i
m
er
g
r
a
p
h
,
,
th
e
n
u
m
b
e
r
o
f
v
e
r
tices
th
at
h
av
e
ec
ce
n
tr
icities
0
,
1
…
=
a
r
e
g
iv
en
b
y
=
1
an
d
−
1
=
(
−
1
)
−
,
=
1
,
2
…
.
Pro
o
f
:
W
e
k
n
o
w
th
at
ev
er
y
p
e
n
d
an
t
v
er
tex
in
,
h
as
(
−
1
)
d
iam
etr
ically
o
p
p
o
s
ite
v
er
tices.
Mo
r
eo
v
er
,
th
e
n
u
m
b
er
o
f
p
en
d
a
n
t
v
e
r
tices
in
,
is
(
−
1
)
−
1
.
T
h
er
ef
o
r
e,
t
h
e
n
u
m
b
e
r
o
f
v
er
tices
t
h
a
t
h
a
v
e
ec
ce
n
tr
ici
ty
0
=
2
is
0
=
(
−
1
)
−
1
.
Similar
ly
,
we
o
b
tain
th
e
n
u
m
b
e
r
o
f
v
e
r
tices
with
ec
ce
n
tr
icities
,
=
1
,
2
…
−
1
as
=
(
−
1
)
−
−
1
,
=
1
,
2
…
−
1
.
Fin
ally
,
th
e
ce
n
tr
e
v
e
r
tex
is
th
e
o
n
ly
v
er
tex
o
f
r
a
d
iu
s
;
h
en
ce
,
we
h
a
v
e:
=
1
an
d
−
1
=
(
−
1
)
−
,
=
1
,
2
…
.
e)
T
h
eo
r
em
3
.
6
:
L
et
A
b
e
a
tr
ee
d
en
d
r
im
e
r
g
r
ap
h
,
,
(
,
>
2
)
,
o
f
o
r
d
er
(
(
−
1
)
−
1
−
2
)
+
1
.
T
h
en
,
t
h
e
r
a
d
io
n
u
m
b
er
o
f
A
s
atis
f
ies
(
)
≥
(
(
−
1
)
−
1
−
2
)
+
+
(
∑
2
(
2
−
(
2
−
)
)
(
−
1
)
−
−
1
−
1
=
1
)
.
Pro
o
f
:
Fro
m
L
em
m
a
3
.
3
,
we
h
av
e:
=
1
;
h
en
ce
,
we
m
u
s
t
ap
p
ly
th
e
s
ec
o
n
d
p
ar
t
o
f
th
e
r
esu
lt
in
T
h
eo
r
em
3
.
5
as f
o
llo
ws:
(
)
≥
−
(
−
)
−
(
−
−
1
)
+
∑
2
(
−
)
=
1
=
(
(
−
1
)
−
1
−
2
)
+
1
−
(
2
−
)
−
(
2
−
(
2
−
1
)
)
+
(
∑
2
(
2
−
(
2
−
)
)
(
−
1
)
−
−
1
−
1
=
1
)
+
2
(
2
−
(
2
−
)
)
=
(
(
−
1
)
−
1
−
2
)
+
+
(
∑
2
(
2
−
(
2
−
)
)
(
−
1
)
−
−
1
−
1
=
1
)
Hen
ce
,
th
e
th
eo
r
em
is
p
r
o
v
en
.
B
y
co
m
b
in
in
g
T
h
eo
r
em
3
.
4
w
ith
T
h
eo
r
em
3
.
6
,
th
e
f
o
llo
win
g
th
eo
r
em
is
o
b
tain
ed
:
f)
T
h
eo
r
em
3
.
7
:
Fo
r
≥
2
−
3
,
th
e
r
ad
i
o
n
u
m
b
er
o
f
tr
ee
d
en
d
r
im
er
g
r
ap
h
,
(
,
>
2
)
s
atis
f
ies
(
(
−
1
)
−
1
−
2
)
+
+
(
∑
2
(
2
−
(
2
−
)
)
(
−
1
)
−
−
1
−
1
=
1
)
≤
(
,
)
≤
+
(
2
−
1
)
+
1
+
∑
(
2
−
1
)
(
(
−
1
)
−
−
1
)
)
(
−
2
)
+
(
2
−
1
)
(
)
−
−
1
−
1
.
−
1
=
1
4.
CO
NCLU
SI
O
N
T
h
e
u
p
p
er
b
o
u
n
d
f
o
r
th
e
(
2
,
1
)
lab
ellin
g
n
u
m
b
e
r
o
f
,
as
3
+
1
f
o
r
,
>
2
h
as b
ee
n
o
b
tain
ed
in
th
is
r
esear
ch
s
tu
d
y
.
T
h
e
u
p
p
er
an
d
lo
wer
b
o
u
n
d
s
h
a
v
e
also
b
e
en
d
eter
m
in
ed
f
o
r
th
e
r
ad
io
lab
ellin
g
f
o
r
,
.
Fo
r
<
2
−
3
,
th
e
r
ad
io
lab
ellin
g
p
r
o
b
lem
f
o
r
,
is
s
till
an
o
p
e
n
p
r
o
b
lem
.
Fu
r
t
h
er
r
esear
ch
ca
n
b
e
e
x
ten
d
e
d
t
o
id
en
tif
y
m
o
r
e
ch
em
ical
s
tr
u
ct
u
r
es
an
d
s
tu
d
y
th
eir
p
r
o
p
e
r
ties
d
u
e
to
th
eir
v
ar
io
u
s
a
p
p
licatio
n
s
in
th
e
f
ield
o
f
co
m
m
u
n
icatio
n
en
g
i
n
ee
r
in
g
.
RE
F
E
R
E
NC
E
S
[
1
]
J.
R
.
G
r
i
g
g
s
a
n
d
R
.
K
.
Y
e
h
,
“
La
b
e
l
l
i
n
g
g
r
a
p
h
s
w
i
t
h
a
c
o
n
d
i
t
i
o
n
a
t
d
i
st
a
n
c
e
2
,
”
S
I
A
M
J
o
u
r
n
a
l
o
n
D
i
sc
ret
e
Ma
t
h
e
m
a
t
i
c
s
,
v
o
l
.
5
,
n
o
.
4
,
p
p
.
5
8
6
-
5
9
5
,
1
9
9
2
,
d
o
i
:
1
0
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B
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m
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m
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n
Za
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p
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Y
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5
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.
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n
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R
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j
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si
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g
h
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k
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r
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w
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sma
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met
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r
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rn
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t
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o
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l
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m
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p
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0
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B
I
O
G
RAP
H
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S O
F
AUTH
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RS
K
in
s
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n
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k
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d
h
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n
d
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fr
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m
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a
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m
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iy
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m
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d
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ra
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r
Un
iv
e
rsity
,
Ti
ru
n
e
lv
e
li
,
Tam
il
Na
d
u
,
I
n
d
ia.
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re
c
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e
d
h
is
M
.
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d
P
h
.
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in
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m
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ti
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s
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ll
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h
e
n
n
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,
I
n
d
ia.
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is
c
u
rre
n
tl
y
wo
r
k
in
g
a
s
a
n
As
sista
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t
P
ro
fe
ss
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r
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th
e
d
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p
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rtme
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t
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f
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y
o
la
C
o
ll
e
g
e
,
Ch
e
n
n
a
i,
I
n
d
ia.
His
a
re
a
s
o
f
re
s
e
a
rc
h
in
g
ra
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h
th
e
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r
y
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re
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h
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n
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l
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ss
ig
n
m
e
n
t
p
r
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lem
s
a
n
d
v
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rio
u
s
g
ra
p
h
lab
e
ll
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n
g
p
ro
b
lem
s.
Alo
n
g
with
1
4
y
e
a
rs
o
f
tea
c
h
in
g
e
x
p
e
rien
c
e
,
h
e
h
a
s
p
lay
e
d
k
e
y
ro
les
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o
r
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n
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n
g
m
a
n
y
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n
tern
a
ti
o
n
a
l
c
o
n
fe
re
n
c
e
s,
se
m
in
a
rs,
a
n
d
wo
r
k
sh
o
p
s.
His
p
a
ss
io
n
i
n
v
o
lv
e
s
m
o
ti
v
a
ti
n
g
th
e
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n
g
m
in
d
s
to
e
x
c
e
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n
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h
e
c
u
rre
n
t
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rc
h
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re
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s
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n
d
e
n
c
o
u
r
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g
in
g
th
e
sta
ff
m
e
m
b
e
rs
to
write
re
se
a
rc
h
a
rti
c
les
th
ro
u
g
h
fa
c
u
lt
y
d
e
v
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l
o
p
m
e
n
t
p
ro
g
ra
m
s.
He
c
a
n
b
e
c
o
n
tac
ted
th
ro
u
g
h
th
is
e
m
a
il
:
jec
in
th
o
k
i
n
s@
re
d
iffma
il
.
c
o
m
Mo
h
a
m
m
e
d
K
.
A.
K
a
a
b
a
r
re
c
e
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e
d
a
ll
h
is
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n
d
e
r
g
ra
d
u
a
te
a
n
d
g
r
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d
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a
te
d
e
g
re
e
s
in
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a
t
h
e
m
a
ti
c
s
fro
m
Was
h
i
n
g
t
o
n
S
tate
Un
i
v
e
rsity
(W
S
U),
P
u
ll
m
a
n
,
WA,
USA.
Ka
a
b
a
r
h
a
s
a
g
l
o
b
a
l
d
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v
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rse
e
x
p
e
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e
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tea
c
h
in
g
a
n
d
h
a
s
wo
r
k
e
d
a
s
a
d
j
u
n
c
t
m
a
th
P
ro
fe
ss
o
r,
m
a
th
lab
in
stru
c
t
o
r,
a
n
d
lec
tu
re
r
a
t
v
a
ri
o
u
s
US
in
stit
u
ti
o
n
s
su
c
h
a
s
M
o
re
n
o
Va
ll
e
y
Co
ll
e
g
e
,
Was
h
i
n
g
t
o
n
S
tate
Un
iv
e
rsity
,
a
n
d
Co
l
o
ra
d
o
E
a
rly
Co
ll
e
g
e
s.
He
a
u
th
o
re
d
two
m
a
th
tex
tb
o
o
k
s,
a
n
d
h
e
is
a
n
i
n
v
it
e
d
re
fe
re
e
f
o
r
m
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re
t
h
a
n
2
0
0
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c
ien
c
e
,
Tec
h
n
o
l
o
g
y
,
En
g
i
n
e
e
rin
g
,
a
n
d
M
a
t
h
e
m
a
ti
c
s
(S
TE
M
)
i
n
tern
a
ti
o
n
a
l
c
o
n
fe
re
n
c
e
s
a
ll
o
v
e
r
t
h
e
wo
rl
d
.
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se
rv
e
d
a
s
a
n
e
d
it
o
r
fo
r
th
e
Am
e
rica
n
M
a
th
e
m
a
ti
c
a
l
S
o
c
iety
(AM
S
)
G
ra
d
u
a
te
S
tu
d
e
n
t
Blo
g
,
a
n
d
h
e
is
a
lso
a
fu
ll
e
d
it
o
r
fo
r
a
n
e
d
u
c
a
ti
o
n
a
l
p
r
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g
ra
m
(M
a
t
h
e
m
a
ti
c
s
S
e
c
ti
o
n
)
a
t
Ca
li
fo
r
n
ia
S
tate
Un
i
v
e
rsity
,
Lo
n
g
Be
a
c
h
,
CA,
USA.
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is
a
n
a
p
p
ro
v
e
d
m
e
m
b
e
r
o
f
S
c
ien
c
e
a
n
d
De
m
o
c
ra
c
y
Ne
two
rk
(S
DN
)
wh
ich
is
a
p
a
rt
o
f
t
h
e
S
c
ien
c
e
,
Tec
h
n
o
lo
g
y
,
a
n
d
S
o
c
iety
p
ro
g
ra
m
a
t
Ha
rv
a
rd
Ke
n
n
e
d
y
S
c
h
o
o
l.
Ka
a
b
a
r
is
a
lso
c
u
rre
n
tl
y
se
rv
i
n
g
a
s
a
n
e
d
it
o
r
fo
r
1
7
i
n
tern
a
ti
o
n
a
l
s
c
ien
ti
fic
jo
u
rn
a
l
s
in
a
p
p
li
e
d
m
a
th
e
m
a
ti
c
s
a
n
d
e
n
g
in
e
e
rin
g
.
He
is also
a
m
e
m
b
e
r
o
f
th
e
e
x
p
e
rt
e
d
i
to
rial
a
d
v
is
o
ry
b
o
a
rd
fo
r
t
h
e
Big
Da
ta
a
n
d
Ad
v
a
n
c
e
d
Wi
re
les
s
Tec
h
n
o
lo
g
ies
S
p
r
in
g
e
r
Bo
o
k
S
e
ries
.
His
re
se
a
r
c
h
in
tere
sts
a
re
fra
c
ti
o
n
a
l
c
a
lcu
lu
s,
a
p
p
li
e
d
m
a
th
e
m
a
t
ics
,
m
a
th
e
m
a
ti
c
a
l
p
h
y
sic
s,
a
n
d
m
a
th
e
d
u
c
a
ti
o
n
.
He
c
a
n
b
e
c
o
n
tac
ted
th
r
o
u
g
h
t
h
is em
a
il
:
m
o
h
a
m
m
e
d
.
k
a
a
b
a
r@ws
u
.
e
d
u
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