I
nte
rna
t
io
na
l J
o
urna
l o
f
E
lect
rica
l a
nd
Co
m
pu
t
er
E
ng
ineering
(
I
J
E
CE
)
Vo
l.
15
,
No
.
5
,
Octo
b
er
20
25
,
p
p
.
4
7
1
4
~
4
7
2
2
I
SS
N:
2088
-
8
7
0
8
,
DOI
: 1
0
.
1
1
5
9
1
/ijece.
v
15
i
5
.
pp
4
7
1
4
-
4
7
2
2
4714
J
o
ur
na
l ho
m
ep
a
g
e
:
h
ttp
:
//ij
ec
e.
ia
esco
r
e.
co
m
Influence
of
the
g
ra
ph density
on
appro
x
ima
te
alg
o
rithms
for the
gra
ph vert
ex
colo
ring
p
ro
blem
Velin K
ra
lev
,
Ra
do
s
la
v
a
K
r
a
lev
a
De
p
a
rtme
n
t
o
f
In
fo
rm
a
ti
c
s,
S
o
u
t
h
-
Wes
t
Un
iv
e
rsit
y
“
Ne
o
fit
Ril
s
k
i
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,
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o
e
v
g
ra
d
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Bu
l
g
a
ria
Art
icle
I
nfo
AB
S
T
RAC
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r
ticle
his
to
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y:
R
ec
eiv
ed
Au
g
1
4
,
2
0
2
4
R
ev
is
ed
Ap
r
1
4
,
2
0
2
5
Acc
ep
ted
J
u
l 3
,
2
0
2
5
Th
is
re
se
a
rc
h
e
x
p
l
o
re
s
two
h
e
u
ris
ti
c
a
lg
o
rit
h
m
s
d
e
sig
n
e
d
t
o
e
fficie
n
tl
y
s
o
lv
e
th
e
g
ra
p
h
c
o
lo
ri
n
g
p
r
o
b
lem
.
Th
e
imp
lem
e
n
tatio
n
c
o
d
e
s
fo
r
b
o
t
h
a
lg
o
rit
h
m
s
a
re
p
ro
v
i
d
e
d
fo
r
b
e
tt
e
r
u
n
d
e
r
sta
n
d
in
g
a
n
d
p
ra
c
ti
c
a
l
a
p
p
li
c
a
t
io
n
.
Th
e
e
x
p
e
rime
n
tal
m
e
th
o
d
o
lo
g
y
is
th
o
ro
u
g
h
ly
d
isc
u
ss
e
d
to
e
n
su
re
c
l
a
rit
y
a
n
d
re
p
ro
d
u
c
ib
il
it
y
.
Th
e
e
x
e
c
u
ti
o
n
ti
m
e
s
o
f
th
e
a
lg
o
rit
h
m
s
we
re
m
e
a
su
re
d
b
y
ru
n
n
in
g
t
h
e
tes
t
a
p
p
li
c
a
ti
o
n
s
six
ti
m
e
s
fo
r
e
a
c
h
a
n
a
ly
z
e
d
g
ra
p
h
.
T
h
e
re
su
lt
s
in
d
ica
te
th
a
t
th
e
first
a
lg
o
rit
h
m
g
e
n
e
ra
ll
y
p
ro
d
u
c
e
d
b
e
tt
e
r
so
l
u
ti
o
n
s
th
a
n
th
e
se
c
o
n
d
.
In
o
n
ly
two
i
n
sta
n
c
e
s
d
id
th
e
first
a
l
g
o
r
it
h
m
p
ro
d
u
c
e
so
l
u
ti
o
n
s
c
o
m
p
a
ra
b
le
to
th
o
se
o
f
th
e
se
c
o
n
d
.
Th
e
re
su
lt
s
re
v
e
a
l
a
n
o
th
e
r
tr
e
n
d
:
a
s
t
h
e
g
ra
p
h
d
e
n
sity
e
x
c
e
e
d
s
8
5
%
,
t
h
e
n
u
m
b
e
r
o
f
re
q
u
ire
d
c
o
lo
rs
in
c
re
a
se
s
sig
n
ifi
c
a
n
t
ly
fo
r
b
o
t
h
a
lg
o
rit
h
m
s
.
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we
v
e
r,
e
v
e
n
a
t
a
d
e
n
sity
o
f
9
5
%
,
th
e
n
u
m
b
e
r
o
f
c
o
lo
rs
re
q
u
ired
to
c
o
l
o
r
t
h
e
g
ra
p
h
'
s
v
e
rti
c
e
s
d
o
e
s
n
o
t
e
x
c
e
e
d
h
a
lf
th
e
to
tal
n
u
m
b
e
r
o
f
v
e
rti
c
e
s.
As
th
e
g
ra
p
h
d
e
n
sit
y
in
c
re
a
se
s
fro
m
9
5
%
to
1
0
0
%
,
t
h
e
n
u
m
b
e
r
o
f
c
o
l
o
rs
re
q
u
ired
to
c
o
l
o
r
th
e
g
ra
p
h
r
ise
s
sig
n
i
fica
n
tl
y
.
Ho
we
v
e
r,
wh
e
n
t
h
e
g
ra
p
h
d
e
n
si
ty
e
x
c
e
e
d
s
9
7
%
,
b
o
th
a
lg
o
rit
h
m
s
p
ro
d
u
c
e
id
e
n
ti
c
a
l
so
l
u
ti
o
n
s.
K
ey
w
o
r
d
s
:
C
h
r
o
m
atic
n
u
m
b
er
Gr
ap
h
co
lo
r
in
g
Gr
ap
h
d
e
n
s
ity
Gr
ap
h
th
eo
r
y
Gr
ee
d
y
alg
o
r
ith
m
T
h
is i
s
a
n
o
p
e
n
a
c
c
e
ss
a
rticle
u
n
d
e
r th
e
CC B
Y
-
SA
li
c
e
n
se
.
C
o
r
r
e
s
p
o
nd
ing
A
uth
o
r
:
Velin
Kr
alev
Dep
ar
tm
en
t o
f
I
n
f
o
r
m
atics,
Facu
lty
o
f
Ma
th
e
m
atics a
n
d
Natu
r
al
Scien
ce
,
So
u
th
-
W
est Un
i
v
er
s
ity
6
6
I
v
a
n
Mic
h
ailo
v
s
tr
.
,
2
7
0
0
B
lag
o
ev
g
r
ad
,
B
u
lg
ar
ia
E
m
ail:
v
elin
_
k
r
alev
@
s
wu
.
b
g
1.
I
NT
RO
D
UCT
I
O
N
I
n
f
o
r
m
ally
,
t
h
e
lab
elin
g
(
o
r
co
lo
r
in
g
)
o
f
a
g
r
a
p
h
'
s
v
er
tices
ca
n
b
e
d
escr
ib
e
d
as
ass
ig
n
in
g
ea
ch
v
e
r
tex
a
s
p
ec
if
ic
lab
el
(
o
r
co
lo
r
)
.
T
h
e
g
o
al
o
f
th
is
p
r
o
ce
s
s
is
to
ass
ig
n
lab
els
(
co
lo
r
s
)
s
u
ch
th
at
n
o
two
ad
jace
n
t
v
er
tices
s
h
ar
e
th
e
s
am
e
lab
el
(
co
lo
r
)
.
T
h
e
la
b
els
(
o
r
c
o
lo
r
s
)
ass
ig
n
ed
to
th
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ar
e
e
lem
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ts
o
f
a
f
i
n
ite
s
et,
with
ea
ch
v
er
tex
r
ec
ei
v
in
g
ex
ac
tly
o
n
e
o
f
th
ese
elem
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n
ts
.
A
co
llectio
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o
f
v
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tices
ass
ig
n
ed
th
e
s
am
e
lab
el
(
co
lo
r
)
is
k
n
o
wn
as
a
ch
r
o
m
at
ic
(
o
r
co
lo
r
)
class
.
I
f
c
-
n
u
m
b
e
r
s
ets
ar
e
f
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r
m
ed
,
i.e
.
,
class
es
ca
n
d
ef
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n
e
a
g
r
ap
h
as
c
-
co
lo
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ab
le.
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r
al
n
u
m
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er
s
f
r
o
m
1
t
h
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o
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g
h
c
ar
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u
s
u
a
lly
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s
ed
to
d
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o
te
th
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c
h
r
o
m
atic
class
es.
Fo
r
a
g
r
ap
h
co
lo
r
in
g
to
b
e
v
alid
,
e
v
er
y
p
air
o
f
ad
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n
t
v
er
tice
s
(
i.e
.
,
v
er
tices
co
n
n
ec
ted
b
y
an
ed
g
e)
m
u
s
t
b
e
ass
ig
n
ed
d
if
f
er
en
t
lab
els
(
co
l
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r
s
)
.
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r
m
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d
i
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en
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al
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it
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e
d
ef
in
ed
th
at
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e
n
a
g
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ap
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co
lo
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ed
ac
ce
p
tab
ly
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co
r
r
ec
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)
f
o
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ex
a
m
p
le
with
c
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lo
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s
,
th
en
it
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c
-
co
lo
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ab
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p
tab
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lo
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o
f
a
g
r
a
p
h
alwa
y
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ex
is
ts
.
I
f
a
g
r
ap
h
h
as
n
v
er
tices,
ea
ch
v
e
r
tex
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n
b
e
ass
ig
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ed
a
u
n
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e
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r
e
s
u
ltin
g
in
ex
ac
tly
n
ch
r
o
m
atic
class
es.
T
h
is
will
ce
r
tain
ly
b
e
an
ac
ce
p
tab
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lab
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lin
g
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r
co
l
o
r
in
g
)
o
f
th
e
g
iv
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n
g
r
a
p
h
.
W
ith
th
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co
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r
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g
s
ch
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e,
n
o
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e
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tices
will
s
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e
s
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,
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[
1
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[
2
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.
T
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s
m
allest
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atic
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
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b
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h
o
f
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en
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p
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f
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ap
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will
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ac
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p
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co
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f
th
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g
r
ap
h
G'
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Mo
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v
er
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e
ch
r
o
m
atic
in
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ex
o
f
a
g
r
ap
h
G'
is
le
s
s
th
an
o
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at
m
o
s
t e
q
u
a
l to
th
e
ch
r
o
m
atic
in
d
e
x
o
f
a
g
r
ap
h
G
[
3
]
,
[
4
]
.
T
h
e
p
r
o
b
lem
o
f
lab
elin
g
(
co
lo
r
in
g
)
v
er
tices
(
in
th
e
f
ield
o
f
g
r
ap
h
th
e
o
r
y
)
is
an
NP
-
h
ar
d
p
r
o
b
lem
[
5
]
an
d
is
s
till
ac
tiv
ely
s
tu
d
ied
[
6
]
–
[
9
]
.
Var
io
u
s
ap
p
r
o
ac
h
es,
m
eth
o
d
s
,
an
d
alg
o
r
ith
m
s
f
o
r
s
o
lv
in
g
th
is
p
r
o
b
lem
co
n
tin
u
e
to
b
e
ac
tiv
ely
r
esear
ch
ed
in
s
cien
tific
liter
atu
r
e.
F
o
r
in
s
tan
ce
,
th
e
m
ax
im
al
in
d
e
p
en
d
en
t
s
et
f
o
r
th
e
v
er
tex
-
co
l
o
r
in
g
p
r
o
b
lem
o
n
p
lan
ar
g
r
ap
h
s
[
1
0
]
,
t
h
e
a
d
jace
n
t
v
er
te
x
-
d
is
tin
g
u
is
h
in
g
ed
g
e
co
lo
r
i
n
g
[
1
1
]
,
t
h
e
r
ain
b
o
w
v
er
tex
c
o
lo
r
in
g
p
r
o
b
lem
[
1
2
]
,
a
n
d
m
a
n
y
o
th
er
s
.
Dif
f
er
en
t
m
eth
o
d
s
o
f
th
e
p
r
o
b
lem
u
s
e
v
a
r
io
u
s
alg
o
r
ith
m
s
[
1
3
]
,
[
1
4
]
,
ap
p
r
o
a
ch
es
[
1
5
]
–
[
1
7
]
a
n
d
tech
n
iq
u
e
s
[
1
8
]
,
[
1
9
]
.
Oth
er
m
et
h
o
d
s
ar
e
u
s
ed
t
o
f
in
d
a
s
o
lu
tio
n
to
s
u
ch
p
r
o
b
lem
s
in
t
h
e
f
ield
o
f
g
r
ap
h
th
eo
r
y
[
2
0
]
.
Ma
n
y
o
th
er
m
o
r
e
d
etailed
an
d
in
-
d
ep
th
an
aly
ze
s
o
f
th
is
p
r
o
b
lem
ar
e
d
is
cu
s
s
ed
in
[
2
1
]
–
[
2
3
]
.
An
y
co
m
p
lete
g
r
ap
h
th
at
d
o
es n
o
t c
o
n
tain
p
ar
allel
ed
g
es a
n
d
lo
o
p
s
ca
n
b
e
co
l
o
r
ed
with
ex
a
ctly
|
V
|
o
f
n
u
m
b
er
o
f
co
lo
r
s
,
wh
er
e
|
V|
i
s
th
e
n
u
m
b
er
o
f
v
er
tices
in
th
i
s
g
r
ap
h
.
T
h
is
is
b
ec
a
u
s
e
in
ev
er
y
co
m
p
lete
g
r
ap
h
f
o
r
ev
e
r
y
p
air
o
f
v
e
r
tices
th
er
e
is
an
ed
g
e
th
at
co
n
n
ec
ts
th
e
s
e
v
er
tices.
Als
o
,
in
ev
er
y
co
m
p
lete
g
r
ap
h
,
ev
er
y
v
er
tex
is
c
o
n
n
ec
te
d
(
b
y
an
ed
g
e)
to
ev
e
r
y
o
th
er
v
er
te
x
.
T
h
e
r
ef
o
r
e,
th
e
ch
r
o
m
atic
in
d
ex
o
f
an
y
c
o
m
p
lete
g
r
ap
h
is
eq
u
al
to
th
e
n
u
m
b
er
o
f
v
er
ti
ce
s
in
th
at
g
r
ap
h
.
B
ec
au
s
e
o
f
t
h
is
s
tatem
en
t,
it
ca
n
b
e
co
n
clu
d
ed
th
at
if
a
g
r
ap
h
h
as
its
co
m
p
lete
s
u
b
g
r
ap
h
,
th
en
th
e
ch
r
o
m
atic
in
d
e
x
o
f
th
i
s
g
r
ap
h
will
b
e
at
least
eq
u
al
(
o
r
g
r
ea
ter
)
to
th
e
n
u
m
b
er
o
f
v
er
tices
th
at
f
o
r
m
th
e
co
m
p
lete
s
u
b
g
r
ap
h
in
th
e
g
iv
en
g
r
a
p
h
.
I
t
is
also
k
n
o
wn
th
at
in
ea
ch
g
r
ap
h
th
at
h
as
its
co
m
p
lete
s
u
b
g
r
ap
h
,
it
is
p
o
s
s
ib
le
f
o
r
th
e
ch
r
o
m
at
ic
in
d
ex
o
f
th
is
g
r
a
p
h
t
o
b
e
g
r
ea
ter
(
b
u
t
n
o
t
less
)
th
an
th
e
n
u
m
b
er
o
f
v
er
tices f
o
r
m
in
g
th
e
c
o
m
p
lete
s
u
b
g
r
ap
h
[
2
4
]
.
I
n
th
is
p
ap
e
r
,
s
o
m
e
r
esu
lts
o
b
tain
ed
af
ter
th
e
ex
ec
u
tio
n
o
f
two
m
o
d
if
ied
v
e
r
s
io
n
s
o
f
t
h
e
g
r
ee
d
y
alg
o
r
ith
m
u
s
ed
to
s
o
lv
e
t
h
e
p
r
o
b
lem
o
f
lab
elin
g
(
co
lo
r
i
n
g
)
th
e
v
e
r
tices
o
f
a
g
r
ap
h
–
Gr
ee
d
y
co
l
o
r
in
g
alg
o
r
ith
m
(
GC
A)
[
2
5
]
ar
e
p
r
e
s
en
ted
an
d
an
aly
ze
d
.
T
h
e
f
ir
s
t
v
er
s
io
n
o
f
th
e
g
r
ee
d
y
alg
o
r
it
h
m
is
clas
s
ical
an
d
th
e
in
itial
ar
r
an
g
e
m
en
t
o
f
th
e
v
er
tices
d
o
es
n
o
t
c
h
an
g
e
wh
en
it
is
ex
ec
u
ted
.
W
ith
th
is
m
o
d
if
icatio
n
,
th
e
v
er
tices
ar
e
lab
eled
(
co
lo
r
e
d
)
in
th
e
o
r
d
er
in
wh
ic
h
th
ey
wer
e
ad
d
ed
to
th
e
g
r
ap
h
-
i.e
.
,
b
y
in
d
e
x
(
g
r
ee
d
y
co
lo
r
in
g
alg
o
r
ith
m
b
ased
o
n
t
h
e
in
d
ex
o
r
d
er
in
g
–
GC
-
I
DX)
.
T
h
e
s
ec
o
n
d
v
a
r
ian
t
o
f
th
e
g
r
ee
d
y
alg
o
r
ith
m
is
im
p
lem
en
ted
as
th
e
in
itial
o
r
d
er
o
f
th
e
v
e
r
tices
is
ch
an
g
ed
,
b
u
t
in
a
r
an
d
o
m
ar
r
an
g
em
e
n
t
way
,
i.e
.
a
r
an
d
o
m
ar
r
an
g
em
e
n
t
o
f
v
er
tices
is
g
en
er
ated
,
o
n
e
f
r
o
m
all
p
o
s
s
ib
le
o
n
es
(
wh
ich
ar
e
ex
ac
tly
n
!
)
.
Af
ter
th
is
s
tep
,
th
e
v
er
tices
o
f
th
e
g
r
a
p
h
ar
e
lab
eled
(
co
lo
r
ed
)
ac
c
o
r
d
i
n
g
to
t
h
e
g
en
e
r
ated
o
r
d
er
o
f
v
er
tice
s
–
i.e
.
,
r
a
n
d
o
m
ly
(
g
r
ee
d
y
c
o
lo
r
in
g
alg
o
r
ith
m
b
a
s
ed
o
n
th
e
r
an
d
o
m
o
r
d
er
in
g
-
GC
-
R
ND)
.
B
o
th
v
ar
ian
ts
o
f
th
e
g
r
ee
d
y
alg
o
r
ith
m
ar
e
ap
p
r
o
x
im
ate,
an
d
th
u
s
,
it
is
n
o
t
g
u
ar
a
n
teed
th
at
eith
er
m
o
d
if
icatio
n
will
f
in
d
th
e
o
p
tim
al
s
o
lu
tio
n
f
o
r
lab
elin
g
(
c
o
lo
r
in
g
)
th
e
g
r
a
p
h
'
s
v
er
tices
u
s
in
g
th
e
f
ewe
s
t
l
ab
els
(
co
lo
r
s
)
p
o
s
s
ib
le.
W
h
e
n
s
o
lv
in
g
NP
-
h
ar
d
p
r
o
b
lem
s
with
lar
g
e
in
p
u
t
d
at
a,
o
n
ly
ap
p
r
o
x
im
ate
alg
o
r
ith
m
s
ca
n
b
e
u
s
ed
,
s
in
ce
th
ey
ca
n
g
en
e
r
ate
at
least
a
clo
s
e
to
th
e
o
p
tim
al
s
o
lu
tio
n
an
d
at
an
ac
ce
p
tab
le
tim
e,
f
o
r
ex
am
p
le
in
th
e
o
r
d
er
o
f
s
ec
o
n
d
s
to
f
ew
m
in
u
tes.
Oth
er
alg
o
r
ith
m
s
f
o
r
s
o
lv
in
g
s
p
ec
if
ic
v
ar
ian
ts
o
f
th
e
g
r
ap
h
v
er
tex
lab
elin
g
(
co
l
o
r
in
g
)
p
r
o
b
l
em
ar
e
also
k
n
o
w
n
an
d
d
is
cu
s
s
ed
in
o
th
e
r
p
u
b
lica
tio
n
s
[
2
6
]
,
[
2
7
]
.
2.
RE
S
E
ARCH
M
E
T
H
O
D
T
h
is
s
ec
tio
n
s
h
o
ws
an
d
d
is
cu
s
s
es
th
e
s
o
u
r
ce
co
d
es
o
f
b
o
th
alg
o
r
ith
m
s
GC
-
I
DX
an
d
GC
-
R
ND
alg
o
r
ith
m
s
.
B
o
th
alg
o
r
ith
m
s
a
r
e
ap
p
r
o
x
im
ate
an
d
s
o
lv
e
th
e
g
r
ap
h
v
er
te
x
c
o
lo
r
in
g
p
r
o
b
lem
ap
p
r
o
x
im
ately
.
Fo
r
b
o
th
alg
o
r
ith
m
s
–
GC
-
I
DX
an
d
GC
-
R
ND,
s
o
m
e
p
ar
am
eter
s
,
d
y
n
a
m
ic
ar
r
a
y
s
an
d
m
atr
ices n
ee
d
to
b
e
d
ec
lar
ed
in
ad
v
an
ce
.
T
h
ey
ar
e
p
r
esen
ted
in
Fig
u
r
e
1
.
01
var
02
│
NumberOfVertices: Integer;
03
│
ArrayOfColors:
array
of
TColor;
04
│
MinimalColors, RandomCount: Integer;
05
│
BestMinimalColors, BestIteration: Integer;
06
│
AdjacencyMatrix:
array
of
array
of
Integer;
Fig
u
r
e
1
.
So
u
r
ce
co
d
e
o
f
p
r
eli
m
in
ar
y
d
ec
la
r
atio
n
s
T
h
e
N
u
mb
erOfVe
r
tices
p
ar
a
m
eter
s
to
r
es
th
e
n
u
m
b
e
r
o
f
v
er
tices
in
th
e
g
r
ap
h
.
T
h
e
A
r
r
a
yOfCo
lo
r
s
ar
r
ay
is
u
s
ed
b
y
b
o
th
alg
o
r
ith
m
s
.
E
ac
h
item
o
f
th
is
d
y
n
am
i
c
s
tr
u
ctu
r
e
co
n
tain
s
an
item
o
f
ty
p
e
TC
o
lo
r
with
wh
ich
th
e
g
iv
en
v
er
tex
o
f
th
e
g
r
ap
h
is
co
lo
r
ed
.
B
o
th
alg
o
r
i
th
m
s
(
C
G
-
I
DX
an
d
C
G
-
R
ND
)
ch
an
g
e
th
e
v
alu
es
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
15
,
No
.
5
,
Octo
b
e
r
20
25
:
4
7
1
4
-
4
7
2
2
4716
o
f
th
ese
item
s
.
T
h
e
Min
ima
l
C
o
lo
r
s
p
ar
am
eter
is
ag
g
r
eg
at
ed
an
d
is
u
s
ed
b
y
t
h
e
alg
o
r
it
h
m
s
in
th
e
s
o
lu
tio
n
s
ea
r
ch
lo
o
p
.
E
ac
h
g
r
ap
h
is
r
e
p
r
esen
ted
b
y
an
ad
jace
n
cy
m
atr
ix
(
lin
e
6
)
,
wh
ich
is
u
s
ed
b
y
b
o
t
h
alg
o
r
ith
m
s
.
E
ac
h
item
[
u
,
v
]
o
f
th
e
m
atr
i
x
s
h
o
ws wh
eth
er
v
er
tices with
in
d
ices u
an
d
v
ar
e
ad
jace
n
t o
r
n
o
t.
T
h
e
Greed
yCo
lo
r
in
g
I
DX
p
r
o
c
ed
u
r
e
im
p
lem
en
ts
th
e
f
ir
s
t
a
p
p
r
o
x
im
ate
m
eth
o
d
f
o
r
lab
elin
g
(
co
lo
r
in
g
)
g
r
ap
h
v
er
tices.
T
h
e
s
o
u
r
ce
c
o
d
e
o
f
th
is
p
r
o
ce
d
u
r
e
is
s
h
o
wn
in
Fig
u
r
e
2
.
T
h
is
p
r
o
ce
d
u
r
e
also
u
s
es
s
o
m
e
ad
d
itio
n
al
p
ar
a
m
eter
s
–
C
u
r
r
en
tC
o
lo
r
,
V
,
V
ertex
an
d
A
cc
ep
t
a
b
leS
o
lu
tio
n
.
T
h
e
C
u
r
r
en
tC
o
l
o
r
p
ar
am
eter
h
o
ld
s
th
e
n
u
m
b
e
r
o
f
o
n
e
o
f
th
e
c
o
lo
r
s
u
s
ed
.
Par
am
eter
V
is
u
s
ed
wh
en
iter
atin
g
th
e
a
d
ja
ce
n
cy
m
atr
ix
wh
e
n
s
ea
r
ch
in
g
f
o
r
th
e
n
eig
h
b
o
r
s
o
f
th
e
g
i
v
en
v
er
tex
.
T
h
e
A
cc
ep
ta
b
leS
o
l
u
tio
n
p
a
r
am
eter
in
d
icate
s
wh
eth
er
th
e
g
iv
en
v
e
r
tex
h
as b
ee
n
s
u
cc
ess
f
u
lly
co
lo
r
e
d
with
an
y
o
f
t
h
e
av
ailab
le
co
lo
r
s
.
01
procedure
GreedyColoringIDX;
02
begin
03
│
var
AcceptableSolution: Boolean := False;
04
│
var
V := 0;
var
Vertex := 0;
var
CurrentColor := 0;
05
│
MinimalColors := 0;
06
│
for
Vertex := 1
to
NumberOfVertices
do
07
│
begin
08
│
│
CurrentColor := 0;
09
│
│
while
not
AcceptableSolution
do
10
│
│
begin
11
│
│
│
CurrentColor := CurrentColor + 1;
12
│
│
│
AcceptableSolution := True;
13
│
│
│
for
V := 1
to
NumberOfVertices
do
14
│
│
│
begin
15
│
│
│
│
if
((AdjacencyMatrix[Vertex][V] > 0)
and
16
│
│
│
│
(ArrayOfColors[V]
=
CurrentColor))
then
17
│
│
│
│
begin
18
│
│
│
│
│
AcceptableSolution := False;
19
│
│
│
│
│
Break;
20
│
│
│
│
end
;
21
│
│
│
end
;
22
│
│
end
;
23
│
│
ArrayOfColors[
Vertex] := CurrentColor;
24
│
│
if
(MinimalColors < CurrentColor)
then
25
│
│
MinimalColors := CurrentColor;
26
│
end
;
27
end
;
Fig
u
r
e
2
.
So
u
r
ce
co
d
e
o
f
th
e
Greed
yCo
lo
r
in
g
I
DX
p
r
o
ce
d
u
r
e
T
h
e
p
ar
am
eter
s
Min
ima
lC
o
lo
r
s
,
V
,
V
ertex
an
d
C
u
r
r
en
tC
o
l
o
r
ar
e
s
et
to
0
in
th
e
b
eg
in
n
i
n
g
o
f
th
e
Greed
yCo
lo
r
in
g
I
DX
p
r
o
ce
d
u
r
e
(
r
o
ws
4
an
d
5
)
.
T
h
e
p
r
o
ce
d
u
r
e
ch
ec
k
s
wh
ich
o
f
th
e
cu
r
r
en
t
co
lo
r
s
ca
n
b
e
u
s
ed
to
co
lo
r
th
e
g
iv
en
v
er
tex
(
th
r
o
u
g
h
a
f
o
r
-
lo
o
p
s
tar
ted
o
n
r
o
w
6
)
.
Fin
d
i
n
g
th
e
co
lo
r
with
t
h
e
s
m
allest
p
o
s
s
ib
le
in
d
ex
to
b
e
u
s
ed
f
o
r
co
l
o
r
in
g
i
s
r
ea
lized
b
y
a
wh
ile
-
lo
o
p
(
r
o
ws
0
9
-
2
2
)
.
J
u
s
t
b
ef
o
r
e
t
h
e
ex
ec
u
tio
n
o
f
th
e
lo
o
p
,
th
e
C
u
r
r
en
t
C
o
lo
r
p
ar
am
eter
is
s
et
to
0
(
r
o
w
0
8
)
.
I
n
lin
e
1
1
,
th
e
C
u
r
r
en
tC
o
lo
r
p
ar
am
eter
is
in
cr
em
en
ted
b
y
1
,
wh
ich
m
ea
n
s
th
at
o
n
th
e
f
ir
s
t
iter
atio
n
o
f
th
e
lo
o
p
,
th
is
p
ar
am
eter
will
b
e
s
et
to
1
.
T
h
r
o
u
g
h
th
e
fo
r
l
o
o
p
(
ex
ec
u
ted
b
etwe
en
r
o
ws
1
1
-
2
1
)
,
it
is
ch
ec
k
ed
w
h
eth
er
a
n
y
o
f
t
h
e
n
eig
h
b
o
r
in
g
(
ad
jac
en
t)
v
er
tices
o
f
th
e
cu
r
r
en
t
v
er
tex
(
p
a
r
am
eter
V
ertex
)
is
n
o
t
alr
ea
d
y
co
lo
r
ed
with
th
e
co
lo
r
C
u
r
r
en
tC
o
lo
r
.
I
f
t
h
is
is
n
o
t
th
e
ca
s
e,
th
en
th
e
cu
r
r
en
t
v
er
tex
is
co
lo
r
ed
with
C
u
r
r
en
tC
o
lo
r
.
T
h
e
ex
ec
u
tio
n
o
f
th
e
fo
r
lo
o
p
ca
n
b
e
p
r
em
atu
r
el
y
ter
m
in
ated
if
a
v
e
r
tex
wh
ich
i
s
ad
jace
n
t
to
t
h
e
cu
r
r
en
t
o
n
e
an
d
is
alr
ea
d
y
co
l
o
r
ed
with
t
h
e
C
u
r
r
en
tC
o
lo
r
is
f
o
u
n
d
.
I
f
th
is
h
ap
p
e
n
s
,
th
e
A
cc
ep
ta
b
leS
o
l
u
tio
n
p
ar
a
m
eter
is
s
et
to
F
a
ls
e
b
ef
o
r
e
th
e
lo
o
p
is
ter
m
in
ated
.
I
n
th
is
s
itu
atio
n
,
th
e
co
lo
r
in
g
o
f
th
e
c
u
r
r
en
t v
e
r
tex
with
th
e
C
u
r
r
en
tC
o
lo
r
is
im
p
o
s
s
ib
le,
an
d
th
e
p
r
o
ce
d
u
r
e
ex
ec
u
tes a
n
ew
iter
atio
n
o
f
th
e
w
h
ile
lo
o
p
an
d
in
cr
ea
s
in
g
t
h
e
in
d
e
x
o
f
th
e
cu
r
r
e
n
t
co
lo
r
b
y
1
(
r
o
w
1
1
)
.
E
x
ec
u
ti
o
n
o
f
th
e
w
h
ile
lo
o
p
(
r
o
ws
0
9
-
2
2
)
c
o
n
t
in
u
es
u
n
til
an
ac
ce
p
ta
b
le
(
p
o
s
s
ib
le)
co
lo
r
is
f
o
u
n
d
f
o
r
th
e
cu
r
r
en
t
v
er
tex
.
I
n
t
h
is
ca
s
e,
th
e
p
ar
am
eter
A
cc
ep
ta
b
l
eS
o
lu
tio
n
will
b
e
s
et
to
Tr
u
e
(
o
n
r
o
w
1
2
b
ef
o
r
e
s
tar
tin
g
th
e
f
o
r
lo
o
p
)
.
T
h
e
v
al
u
e
o
f
th
e
C
u
r
r
en
tC
o
lo
r
p
ar
am
eter
is
s
to
r
ed
in
th
e
ar
r
ay
A
r
r
a
yOfCo
lo
r
s
in
th
e
item
a
s
s
o
ciate
d
with
th
e
p
ar
am
eter
V
ertex
(
r
o
w
2
3
)
.
T
h
e
v
alu
e
o
f
th
e
C
u
r
r
en
tC
o
lo
r
p
ar
am
eter
is
co
p
ied
to
th
e
Min
ima
lC
o
lo
r
s
p
ar
am
eter
o
n
ly
if
th
e
v
alu
e
o
f
th
e
C
u
r
r
en
tC
o
lo
r
p
ar
am
eter
is
g
r
ea
ter
t
h
an
th
e
v
alu
e
o
f
th
e
Min
ima
lC
o
lo
r
s
p
ar
am
eter
.
W
h
en
th
e
v
alu
e
o
f
th
e
Min
ima
lC
o
lo
r
s
p
ar
am
eter
is
in
c
r
e
ased
b
y
o
n
e,
it
m
ea
n
s
t
h
at
th
e
av
ailab
le
co
lo
r
s
ar
e
n
o
t
e
n
o
u
g
h
to
co
l
o
r
th
e
cu
r
r
en
t
v
e
r
tex
an
d
a
n
ew
co
lo
r
n
ee
d
s
to
b
e
a
d
d
ed
,
wh
ich
is
d
o
n
e
b
y
in
cr
ea
s
in
g
b
y
o
n
e
th
e
v
alu
e
o
f
th
e
Min
ima
lC
o
lo
r
s
p
ar
am
eter
.
T
h
e
ex
ec
u
tio
n
o
f
th
e
fo
r
lo
o
p
(
lin
es
0
6
–
2
6
)
en
d
s
o
n
ly
wh
en
all
th
e
v
er
tic
es
in
th
e
g
r
ap
h
ar
e
co
lo
r
e
d
,
a
n
d
in
s
u
ch
a
way
th
at
n
o
p
air
o
f
ad
jace
n
t
v
er
tices
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with
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Af
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r
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r
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eter
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in
im
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m
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ith
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u
cted
in
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way
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T
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Greed
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N
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p
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im
p
lem
en
ts
th
e
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o
n
d
ap
p
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im
ate
m
eth
o
d
f
o
r
co
l
o
r
in
g
g
r
ap
h
v
er
tices.
T
h
e
s
o
u
r
ce
c
o
d
e
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f
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is
p
r
o
ce
d
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r
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Fig
u
r
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3
.
T
h
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p
r
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ce
d
u
r
e
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u
s
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s
o
m
e
ad
d
itio
n
al
p
ar
am
eter
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ima
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o
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s
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estItera
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d
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a
n
d
o
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o
u
n
t
.
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h
e
B
estM
in
ima
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l
o
r
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ar
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eter
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o
ld
s
th
e
m
in
im
al
n
u
m
b
er
o
f
c
o
lo
r
s
n
ee
d
ed
to
co
lo
r
th
e
g
r
ap
h
v
er
t
ices.
T
h
is
p
ar
am
eter
h
as
a
d
if
f
er
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t
p
u
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p
o
s
e
th
a
n
th
e
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ima
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eter
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ich
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e
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im
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b
e
r
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f
co
l
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r
eq
u
ir
e
d
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en
r
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n
n
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th
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r
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t
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n
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n
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Greed
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r
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n
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n
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ast
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ar
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ima
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ar
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eter
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tain
s
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n
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m
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er
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s
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lo
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h
af
ter
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er
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lls
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eter
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ed
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eter
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itialized
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r
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o
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ly
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eter
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ich
ca
ll
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Greed
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est s
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o
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ased
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ich
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e
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a
n
d
o
m
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r
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o
f
th
e
v
er
tices
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g
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ated
b
y
th
e
f
o
r
lo
o
p
(
lin
e
s
0
7
–
1
2
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s
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tin
g
f
r
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m
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h
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er
tex
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o
r
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h
ich
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n
ew
in
d
e
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o
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en
(
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d
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with
w
h
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th
e
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r
r
en
t
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is
ex
ch
a
n
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ed
(
lin
e
1
1
)
.
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n
t
h
e
n
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t
s
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ep
,
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ew
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lt
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ate
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tex
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w
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g
ed
with
s
o
m
e
v
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tex
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r
o
m
th
o
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it.
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p
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s
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n
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all
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p
to
th
e
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ir
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t
h
a
v
e
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ee
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d
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m
ly
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is
g
en
er
ated
,
th
e
p
r
o
ce
d
u
r
e
Greed
yCo
lo
r
in
g
I
DX
is
ca
lled
,
w
h
ich
co
l
o
r
s
th
e
v
er
tices
o
f
th
e
g
r
ap
h
i
n
th
e
th
u
s
g
en
er
ated
s
u
b
o
r
d
er
.
W
h
en
th
e
Greed
yCo
lo
r
in
g
I
DX
p
r
o
ce
d
u
r
e
is
ex
ec
u
ted
,
th
e
n
u
m
b
er
o
f
co
lo
r
s
n
ee
d
e
d
to
co
lo
r
all
v
er
tices
o
f
th
e
g
r
ap
h
is
ca
lcu
lated
.
T
h
is
co
u
n
t
is
s
to
r
ed
in
th
e
Min
ima
lC
o
lo
r
s
p
ar
am
eter
,
wh
ich
is
co
m
p
ar
ed
to
th
e
b
est s
o
lu
tio
n
f
o
u
n
d
f
r
o
m
a
p
r
e
v
io
u
s
iter
atio
n
o
f
th
e
alg
o
r
ith
m
.
I
f
th
e
cu
r
r
e
n
t so
lu
tio
n
is
b
etter
(
i.e
.
th
e
Min
ima
lC
o
lo
r
s
p
ar
a
m
eter
h
as
a
s
m
aller
v
alu
e
t
h
an
th
e
B
estM
in
ima
lC
o
lo
r
s
p
ar
am
eter
)
th
en
th
e
cu
r
r
en
t
s
o
lu
tio
n
is
s
to
r
ed
as
th
e
b
est
f
o
u
n
d
s
o
f
a
r
.
Acc
o
r
d
in
g
l
y
,
th
e
B
estItera
tio
n
p
ar
am
eter
s
to
r
es
th
e
iter
atio
n
at
wh
ich
th
is
(
b
est)
s
o
lu
tio
n
was f
o
u
n
d
.
01
procedure
GreedyColoringRND;
02
begin
03
│
BestMinimalColors := MaxInt;
RandomCount := 100;
04
│
BestIteration := 0;
05
│
for var
Iteration := 1
to
RandomCount
do
06
│
begin
07
│
│
for var
V := NumberOfVertices
downto
1
do
08
│
│
begin
09
│
│
│
var
I := V;
10
│
│
│
var
J := Random(V) + 1;
11
│
│
│
var
T :
= I; I := J; J := T;
// Swap vertices I and J
12
│
│
end
;
13
│
│
GreedyColoringIDX();
14
│
│
if
MinimalColors < BestMinimalColors
then
15
│
│
begin
16
│
│
│
BestMinimalColors := MinimalColors;
17
│
│
│
BestIteration := Iteration;
18
│
│
end
;
19
│
end
;
20
end
;
Fig
u
r
e
3
.
So
u
r
ce
co
d
e
o
f
th
e
Greed
yCo
lo
r
in
g
R
N
D
p
r
o
ce
d
u
r
e
3.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
T
h
e
r
esu
lts
o
f
th
e
ex
p
e
r
im
en
t
will
b
e
p
r
esen
ted
.
An
an
aly
s
is
b
etwe
en
th
e
two
ap
p
r
o
x
im
ate
m
eth
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d
s
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r
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te
d
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ter
m
s
o
f
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e
n
u
m
b
er
o
f
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in
im
u
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co
lo
r
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r
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u
ir
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c
o
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r
th
e
te
s
t
g
r
ap
h
s
,
as
well
as
th
e
r
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n
n
in
g
tim
e
o
f
th
e
alg
o
r
ith
m
s
.
Fo
r
th
is
s
tu
d
y
,
2
4
g
r
a
p
h
s
with
1
0
0
0
v
e
r
tices
an
d
d
if
f
er
en
t
n
u
m
b
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f
ed
g
es
wer
e
cr
ea
ted
.
T
h
ese
g
r
ap
h
s
ar
e
u
n
d
ir
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ted
a
n
d
co
n
tain
n
o
m
u
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es
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p
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r
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T
h
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n
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eter
m
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y
th
e
g
r
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d
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(
in
p
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n
tag
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th
at
is
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et
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r
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ch
g
r
ap
h
.
T
h
is
p
ar
am
eter
r
a
n
g
es
f
r
o
m
5
to
9
5
f
o
r
g
r
a
p
h
s
G1
–
G1
9
an
d
c
h
an
g
es
in
5
p
er
c
en
t
in
cr
em
en
ts
.
Fo
r
g
r
ap
h
s
G2
0
–
G2
4
th
is
p
ar
am
e
ter
r
an
g
es f
r
o
m
9
6
to
1
0
0
a
n
d
ch
an
g
es in
1
p
er
ce
n
t in
cr
em
en
ts
.
Evaluation Warning : The document was created with Spire.PDF for Python.
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OS
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d
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ig
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s
s
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r
: I
n
tel
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T
M
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at
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GHz
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SS
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1
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T
ab
le
1
.
R
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lts
o
f
th
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ap
p
r
o
x
im
ate
alg
o
r
ith
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s
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r
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h
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D
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C
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s
Ti
me
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ms)
C
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4
2
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25
47
31
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1
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3
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_
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15
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4
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35
63
41
2
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42
6
4
G
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9
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20
9
9
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0
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45
76
53
2
6
7
2
1
3
1
8
5
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25
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55
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11
6
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7
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94
10
T
ab
le
2
.
R
esu
lts
o
f
th
e
ap
p
r
o
x
im
ate
alg
o
r
ith
m
s
f
o
r
g
r
a
p
h
s
G
2
0
–
G2
4
G
r
a
p
h
n
u
m
b
e
r
G
r
a
p
h
D
e
n
si
t
y
Ed
g
e
s
G
r
e
e
d
y
C
o
l
o
r
i
n
g
(
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D
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e
d
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l
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r
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g
(
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f
f
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f
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n
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me
(
%)
(
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n
t
)
C
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l
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s
Ti
me
(
ms)
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s
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me
(
ms)
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t
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a
t
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c
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s
20
G
_
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4
7
9
5
2
0
96
4
7
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5
2
0
4
1
9
2
6
5
4
2
2
2
9
7
5
65
3
21
G
_
1
0
0
0
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4
8
4
5
1
6
97
4
8
4
5
1
6
4
6
1
2
8
7
4
6
0
3
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5
2
27
1
22
G
_
1
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0
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4
8
9
5
1
0
98
4
8
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5
6
3
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5
6
3
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41
0
23
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_
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24
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_
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79
0
I
n
T
ab
le
s
1
an
d
2
,
th
e
“
Gr
ap
h
n
u
m
b
er
”
co
lu
m
n
s
h
o
ws
th
e
n
u
m
b
er
o
f
th
e
test
g
r
ap
h
.
T
h
e
“
Gr
ap
h
f
ile
n
am
e
”
co
lu
m
n
s
h
o
ws
th
e
n
am
e
o
f
th
e
f
ile
wh
er
e
th
e
in
f
o
r
m
atio
n
f
o
r
th
e
co
r
r
esp
o
n
d
in
g
test
g
r
ap
h
is
s
to
r
ed
.
T
h
e
“
Den
s
ity
(
%)
”
co
lu
m
n
s
h
o
ws th
e
d
en
s
ity
o
f
th
e
test
g
r
ap
h
in
ter
m
s
o
f
th
e
n
u
m
b
er
o
f
ed
g
es th
at
th
is
g
r
ap
h
co
n
tain
s
o
u
t
o
f
all
th
e
p
o
s
s
ib
le
ed
g
es
t
h
at
th
is
g
r
ap
h
wo
u
ld
h
av
e
if
it
wer
e
co
m
p
lete.
Acc
o
r
d
in
g
ly
,
th
e
co
lu
m
n
“
E
d
g
e
c
o
u
n
t
”
s
h
o
ws
th
e
n
u
m
b
er
o
f
th
ese
e
d
g
es.
T
h
e
“
C
o
l
o
r
”
co
l
u
m
n
s
s
h
o
w
th
e
n
u
m
b
e
r
o
f
r
eq
u
ir
ed
co
lo
r
s
th
at
ea
ch
o
f
th
e
alg
o
r
ith
m
s
u
s
ed
to
co
lo
r
th
e
co
r
r
esp
o
n
d
in
g
g
r
ap
h
.
Acc
o
r
d
in
g
l
y
,
th
e
“
T
im
e
(
m
s
)
”
co
lu
m
n
s
s
h
o
w
th
e
tim
e
f
o
r
ea
c
h
o
f
th
e
two
alg
o
r
ith
m
s
to
g
e
n
er
ate
a
s
o
lu
tio
n
.
T
h
e
“
I
ter
atio
n
”
co
l
u
m
n
s
h
o
ws
th
e
b
est
iter
atio
n
o
f
t
h
e
GC
-
R
ND
alg
o
r
ith
m
wh
er
e
th
e
alg
o
r
ith
m
g
en
er
ated
t
h
e
b
est
s
o
lu
tio
n
o
u
t
o
f
a
to
tal
o
f
2
5
0
iter
atio
n
s
p
er
f
o
r
m
ed
.
T
h
e
“
Dif
f
in
c
o
lo
r
s
”
c
o
lu
m
n
s
h
o
ws
th
e
d
if
f
er
en
ce
in
c
h
r
o
m
atic
class
es
b
etwe
en
th
e
GC
-
R
ND
alg
o
r
ith
m
an
d
th
e
C
G
-
I
DX
alg
o
r
ith
m
.
T
ab
le
1
an
d
th
e
ch
a
r
ts
in
Fig
u
r
e
s
4
an
d
5
s
h
o
w
th
e
r
esu
lt
s
o
f
th
e
two
alg
o
r
ith
m
s
in
ter
m
s
o
f
th
e
n
u
m
b
er
o
f
r
eq
u
ir
e
d
co
lo
r
s
th
at
th
e
two
alg
o
r
ith
m
s
u
s
ed
to
co
lo
r
th
e
g
r
ap
h
s
f
r
o
m
G1
th
r
o
u
g
h
G1
9
.
T
h
e
r
esu
lts
also
s
h
o
w
th
at
f
o
r
all
n
in
etee
n
g
r
ap
h
s
f
r
o
m
G1
th
r
o
u
g
h
G
1
9
,
th
e
GC
-
I
DX
alg
o
r
ith
m
f
o
u
n
d
b
etter
s
o
lu
tio
n
s
th
an
th
e
GC
-
R
ND
alg
o
r
ith
m
.
I
n
two
ca
s
es,
o
n
l
y
th
e
n
u
m
b
e
r
o
f
ch
r
o
m
atic
class
es
(
co
lo
r
s
)
g
e
n
er
ated
b
y
th
e
GC
-
R
ND
alg
o
r
ith
m
ar
e
clo
s
e
to
th
o
s
e
g
en
e
r
ated
b
y
th
e
G
C
-
I
DX
alg
o
r
ith
m
–
th
ese
a
r
e
th
e
ca
s
es
f
o
r
g
r
ap
h
s
G1
4
an
d
G1
7
.
Fo
r
g
r
a
p
h
G1
4
,
th
e
GC
-
I
DX
alg
o
r
ith
m
g
en
er
ated
a
s
o
lu
tio
n
wh
e
r
e
1
7
4
co
l
o
r
s
wer
e
n
ee
d
e
d
to
co
lo
r
th
e
g
r
ap
h
.
Acc
o
r
d
in
g
l
y
,
th
e
GC
-
R
ND
alg
o
r
ith
m
f
o
u
n
d
a
v
er
y
clo
s
e
s
o
lu
tio
n
,
wh
er
e
1
7
5
co
lo
r
s
wer
e
n
ee
d
ed
to
c
o
lo
r
th
e
g
r
ap
h
(
i.
e.
,
o
n
l
y
o
n
e
c
o
lo
r
m
o
r
e
th
a
n
th
e
co
lo
r
s
t
h
e
GC
-
I
DX
alg
o
r
ith
m
u
s
ed
)
.
I
n
all
o
th
er
ca
s
es,
th
e
GC
-
I
DX
alg
o
r
ith
m
f
o
u
n
d
s
o
lu
tio
n
s
th
at
wer
e
b
etter
th
an
th
o
s
e
f
o
u
n
d
b
y
t
h
e
G
C
-
R
ND
alg
o
r
ith
m
.
T
h
e
d
if
f
er
e
n
ce
in
th
e
n
u
m
b
er
o
f
co
lo
r
s
v
ar
ies
b
etwe
en
4
an
d
1
2
,
with
g
r
ap
h
s
G1
an
d
G1
6
h
av
in
g
a
d
if
f
er
en
ce
o
f
4
co
l
o
r
s
an
d
g
r
ap
h
G
1
3
h
a
v
in
g
a
d
if
f
er
en
ce
o
f
1
2
co
lo
r
s
as
s
h
o
wn
in
Fig
u
r
e
5
.
H
o
wev
er
,
Fig
u
r
e
s
4
an
d
5
s
h
o
w
th
at
as
th
e
n
u
m
b
er
o
f
ed
g
es
in
th
e
g
r
ap
h
in
cr
ea
s
es,
wh
ich
is
a
co
n
s
eq
u
en
ce
o
f
th
e
g
r
ap
h
d
e
n
s
ity
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
7
0
8
I
n
flu
en
ce
o
f th
e
g
r
a
p
h
d
e
n
s
ity
o
n
a
p
p
r
o
xima
te
a
lg
o
r
ith
ms fo
r
…
(
V
elin
K
r
a
lev
)
4719
in
cr
ea
s
in
g
,
th
e
GC
-
R
ND
alg
o
r
ith
m
b
eg
i
n
s
to
g
en
er
ate
s
o
lu
tio
n
s
th
at
a
r
e
clo
s
er
to
th
o
s
e
g
e
n
er
ated
b
y
GC
-
I
DX
alg
o
r
ith
m
.
Fr
o
m
th
e
tr
en
d
o
f
th
e
o
b
tain
e
d
r
esu
lts
,
it
ca
n
b
e
co
n
clu
d
ed
th
at
f
o
r
g
r
a
p
h
s
wi
th
a
h
ig
h
er
d
en
s
ity
,
f
o
r
ex
a
m
p
le
g
r
ea
te
r
th
an
7
0
%,
th
e
GC
-
R
ND
alg
o
r
ith
m
ca
n
b
e
s
u
cc
ess
f
u
lly
u
s
ed
to
g
en
e
r
at
e
s
o
lu
tio
n
s
.
Fig
u
r
e
4
.
T
h
e
n
u
m
b
er
o
f
c
o
lo
r
s
(
lef
t y
-
ax
is
)
g
e
n
er
ated
b
y
th
e
two
alg
o
r
ith
m
s
at
d
if
f
er
en
t
g
r
ap
h
d
e
n
s
ities
(
r
ig
h
t y
-
ax
is
)
Fig
u
r
e
5
.
Dif
f
e
r
en
ce
in
ch
r
o
m
atic
class
e
s
b
etwe
en
th
e
GC
(
R
ND)
alg
o
r
ith
m
an
d
t
h
e
C
G
(
I
DX)
alg
o
r
ith
m
T
h
e
ch
ar
t
in
Fig
u
r
e
4
s
h
o
ws
a
n
o
th
er
t
r
en
d
,
wh
ich
is
r
elate
d
to
th
e
n
u
m
b
e
r
o
f
r
e
q
u
ir
ed
co
l
o
r
s
th
at
th
e
two
alg
o
r
ith
m
s
u
s
ed
in
f
in
d
in
g
a
s
o
lu
tio
n
.
W
h
en
i
n
cr
ea
s
in
g
th
e
d
en
s
ity
o
f
t
h
e
g
r
a
p
h
,
f
o
r
ex
am
p
le
f
o
r
v
alu
es
g
r
ea
ter
th
an
8
5
%,
th
e
n
u
m
b
er
o
f
r
eq
u
i
r
ed
co
lo
r
s
in
cr
ea
s
es
s
ig
n
if
ican
tly
,
an
d
th
is
is
tr
u
e
f
o
r
b
o
th
alg
o
r
ith
m
s
-
GC
-
I
DX
alg
o
r
ith
m
an
d
GC
-
R
ND
alg
o
r
ith
m
.
Ho
wev
e
r
,
th
e
n
u
m
b
er
o
f
c
o
lo
r
s
n
ee
d
e
d
to
co
lo
r
th
e
v
er
tices
o
f
a
g
r
ap
h
,
ev
e
n
at
a
d
en
s
ity
o
f
9
5
%
d
o
es
n
o
t
ex
ce
ed
m
o
r
e
th
a
n
h
alf
th
e
n
u
m
b
e
r
o
f
v
er
tices
in
th
e
s
am
e
g
r
ap
h
,
s
in
ce
3
9
4
/1
0
0
0
=
0
.
3
9
4
.
T
h
is
is
an
im
p
o
r
tan
t
r
esu
lt
o
f
h
o
w
in
cr
ea
s
in
g
th
e
d
en
s
ity
o
f
th
e
g
r
ap
h
af
f
ec
ts
th
e
n
u
m
b
er
o
f
ch
r
o
m
atic
class
es
g
en
er
ated
b
y
th
e
alg
o
r
ith
m
s
.
I
t
is
k
n
o
wn
th
at
f
o
r
a
co
m
p
lete
g
r
ap
h
,
i.e
.
,
at
1
0
0
%
d
en
s
ity
,
th
e
n
u
m
b
er
o
f
c
o
lo
r
s
r
eq
u
ir
e
d
to
co
l
o
r
a
g
r
ap
h
eq
u
al
to
th
e
n
u
m
b
e
r
o
f
v
er
tices
o
f
th
at
g
r
ap
h
.
T
h
is
m
ea
n
s
th
at
wh
e
n
t
h
e
d
en
s
ity
o
f
th
e
g
r
ap
h
ch
an
g
es
b
etwe
en
9
5
%
a
n
d
1
0
0
%,
th
e
n
u
m
b
er
o
f
c
o
lo
r
s
n
ee
d
ed
to
co
lo
r
th
e
g
r
a
p
h
i
n
cr
ea
s
es
s
ig
n
if
ican
tly
.
T
h
is
is
ex
ac
tly
wh
at
th
e
r
esu
lts
p
r
esen
ted
in
T
ab
l
e
2
.
I
t
ca
n
also
b
e
s
ee
n
th
at
at
h
ig
h
v
al
u
es
o
f
t
h
e
g
r
ap
h
d
e
n
s
ity
p
ar
am
eter
,
b
o
t
h
alg
o
r
ith
m
s
g
en
e
r
ate
id
en
tica
l
s
o
lu
tio
n
s
,
s
u
ch
as
th
e
r
esu
lts
o
b
tain
ed
f
o
r
g
r
ap
h
s
with
d
en
s
ity
ab
o
v
e
9
7
%.
T
h
e
ch
ar
t
in
Fig
u
r
e
6
illu
s
tr
ates
h
o
w
in
cr
ea
s
in
g
th
e
n
u
m
b
e
r
o
f
ed
g
es
(
an
d
th
u
s
th
e
g
r
ap
h
d
en
s
ity
)
im
p
ac
ts
th
e
ex
ec
u
tio
n
tim
e
o
f
th
e
two
alg
o
r
ith
m
s
.
T
h
e
ex
ec
u
tio
n
tim
e
o
f
th
e
GC
-
R
ND
alg
o
r
ith
m
is
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
15
,
No
.
5
,
Octo
b
e
r
20
25
:
4
7
1
4
-
4
7
2
2
4720
s
ig
n
if
ican
tly
lo
n
g
er
th
an
th
at
o
f
th
e
GC
-
I
DX
alg
o
r
ith
m
.
T
h
is
d
if
f
er
e
n
ce
v
ar
ies:
f
o
r
e
x
a
m
p
le,
in
g
r
ap
h
G1
,
wh
ich
h
as
t
h
e
lo
west
d
en
s
ity
,
th
e
d
if
f
e
r
en
ce
is
th
e
la
r
g
est,
w
h
ile
in
g
r
ap
h
G1
9
,
w
h
ich
h
as
t
h
e
h
ig
h
est
d
en
s
ity
,
th
e
d
if
f
e
r
en
ce
is
th
e
s
m
allest.
Ho
wev
er
,
f
o
r
ea
ch
r
u
n
o
f
th
e
GC
-
R
ND
alg
o
r
ith
m
,
1
0
0
iter
a
tio
n
s
ar
e
p
er
f
o
r
m
ed
,
an
d
th
e
iter
atio
n
th
at
g
en
er
at
es
th
e
b
est
s
o
lu
tio
n
is
s
elec
t
ed
.
T
h
e
tim
es
p
r
esen
ted
in
T
ab
le
1
in
clu
d
e
th
e
ex
ec
u
tio
n
tim
e
o
f
all
1
0
0
iter
a
tio
n
s
.
T
h
is
m
ea
n
s
th
at
th
e
ac
tu
al
tim
e
f
o
r
a
s
in
g
le
ex
ec
u
tio
n
o
f
th
e
alg
o
r
ith
m
is
,
o
n
av
er
a
g
e,
1
0
0
tim
es
less
.
T
h
e
co
m
p
lex
ity
o
f
b
o
th
alg
o
r
ith
m
s
is
q
u
ad
r
atic,
p
r
im
ar
ily
d
ep
en
d
i
n
g
o
n
th
e
n
u
m
b
er
o
f
v
e
r
tices
in
th
e
g
r
ap
h
,
an
d
to
a
less
er
ex
ten
t
o
n
its
d
en
s
ity
,
i.e
.
,
th
e
n
u
m
b
e
r
o
f
ed
g
es.
T
h
is
is
illu
s
tr
ated
in
Fig
u
r
e
6
,
wh
er
e
a
s
ig
n
if
ican
t
in
cr
ea
s
e
in
t
h
e
n
u
m
b
e
r
o
f
ed
g
es,
a
n
d
th
u
s
th
e
g
r
ap
h
'
s
d
en
s
ity
,
r
esu
lts
in
th
e
ex
ec
u
tio
n
tim
e
o
f
b
o
th
alg
o
r
ith
m
s
ch
an
g
in
g
in
s
ig
n
if
ican
tly
an
d
lin
ea
r
ly
,
wit
h
o
n
ly
a
v
er
y
s
m
all
in
cr
em
en
t.
Fig
u
r
e
6
.
C
o
m
p
a
r
is
o
n
o
f
th
e
e
x
ec
u
tio
n
tim
es o
f
b
o
th
alg
o
r
it
h
m
s
4.
CO
NCLU
SI
O
N
T
h
is
p
ap
er
d
is
cu
s
s
es
a
s
tu
d
y
r
elate
d
t
o
th
e
g
r
a
p
h
lab
elin
g
(
co
l
o
r
in
g
)
p
r
o
b
lem
.
Var
io
u
s
s
cien
tific
an
aly
s
es,
m
eth
o
d
s
,
an
d
alg
o
r
i
th
m
s
r
elate
d
to
th
e
g
r
ap
h
lab
elin
g
(
co
lo
r
in
g
)
p
r
o
b
lem
wer
e
d
is
cu
s
s
ed
.
I
n
th
i
s
p
ap
er
,
two
ap
p
r
o
x
im
ate
alg
o
r
ith
m
s
f
o
r
s
o
lv
in
g
th
e
g
r
ap
h
v
er
tex
co
lo
r
i
n
g
p
r
o
b
lem
wer
e
im
p
lem
en
ted
an
d
p
r
esen
ted
.
T
h
e
d
ec
lar
atio
n
s
o
f
th
e
b
asic
d
ata
s
tr
u
ctu
r
es
u
s
ed
b
y
th
e
alg
o
r
ith
m
s
,
in
cl
u
d
i
n
g
o
n
e
-
d
im
e
n
s
io
n
al
an
d
two
-
d
im
en
s
io
n
al
ar
r
ay
s
,
wer
e
also
p
r
esen
ted
.
T
h
e
p
r
o
g
r
am
co
d
es
o
f
b
o
th
h
e
u
r
is
tic
alg
o
r
ith
m
s
wer
e
p
r
esen
ted
an
d
an
al
y
ze
d
in
d
et
ail.
W
h
en
co
n
d
u
ctin
g
th
e
e
x
p
er
im
en
ts
,
th
e
o
p
er
atin
g
s
y
s
te
m
'
s
ab
il
ity
to
wo
r
k
in
m
u
ltit
ask
in
g
m
o
d
e
was
s
p
ec
if
ically
co
n
s
id
er
ed
.
Acc
o
r
d
in
g
ly
,
s
ix
r
u
n
s
o
f
b
o
t
h
alg
o
r
ith
m
s
wer
e
co
n
d
u
cted
f
o
r
ea
ch
o
f
th
e
2
4
a
n
aly
ze
d
g
r
a
p
h
s
.
T
h
e
a
v
er
ag
e
ex
ec
u
tio
n
tim
es
f
r
o
m
th
ese
r
u
n
s
wer
e
ca
lcu
l
ated
an
d
p
r
esen
ted
in
T
ab
le
s
1
a
n
d
2
.
Fo
r
t
h
e
s
o
lu
tio
n
s
g
en
er
ate
d
b
y
th
e
GC
-
I
D
X
alg
o
r
ith
m
,
id
e
n
tical
r
esu
lts
wer
e
o
b
tain
e
d
i
n
all
r
u
n
s
.
T
h
ese
r
esu
lts
ar
e
al
s
o
p
r
esen
ted
in
T
ab
le
1
an
d
T
ab
le
2
.
I
n
co
n
tr
ast,
th
e
GC
-
R
ND
a
lg
o
r
ith
m
p
r
o
d
u
ce
d
d
if
f
er
en
t
r
esu
lts
ac
r
o
s
s
th
e
s
i
x
r
u
n
s
.
T
ab
le
1
an
d
T
ab
le
2
p
r
esen
t
th
e
b
est
r
esu
lts
g
en
er
ated
f
o
r
ea
ch
g
r
ap
h
ac
r
o
s
s
all
r
u
n
s
.
T
h
e
r
esu
lts
o
f
t
h
is
ex
p
er
im
e
n
t
s
h
o
w
th
at,
f
o
r
g
r
ap
h
s
G1
–
G1
9
,
th
e
GC
-
I
DX
alg
o
r
ith
m
g
en
er
ated
b
etter
s
o
lu
tio
n
s
th
a
n
th
o
s
e
g
en
er
ated
b
y
th
e
GC
-
R
ND
alg
o
r
ith
m
in
m
o
s
t
ca
s
es.
I
n
o
n
ly
two
ca
s
es
d
id
th
e
GC
-
R
ND
alg
o
r
ith
m
g
en
e
r
ate
s
o
lu
tio
n
s
th
at
wer
e
co
m
p
ar
a
b
le
to
th
o
s
e
p
r
o
d
u
ce
d
b
y
th
e
GC
-
I
DX
alg
o
r
ith
m
.
T
h
e
r
esu
lts
in
d
icate
an
o
th
er
tr
en
d
:
as
th
e
g
r
ap
h
d
en
s
ity
ex
ce
ed
s
8
5
%,
th
e
n
u
m
b
er
o
f
r
eq
u
ir
ed
co
lo
r
s
in
cr
ea
s
es
s
ig
n
if
ican
tly
f
o
r
b
o
th
alg
o
r
ith
m
s
.
Ho
wev
er
,
ev
en
at
a
d
e
n
s
ity
o
f
9
5
%,
th
e
n
u
m
b
er
o
f
co
l
o
r
s
r
eq
u
ir
e
d
t
o
co
l
o
r
th
e
v
er
tices
o
f
a
g
r
ap
h
d
o
es
n
o
t
ex
ce
ed
h
alf
th
e
n
u
m
b
er
o
f
v
er
tices
in
th
e
g
r
ap
h
.
T
h
is
is
an
im
p
o
r
ta
n
t
f
in
d
in
g
th
at
d
em
o
n
s
tr
ates
h
o
w
in
cr
ea
s
in
g
th
e
g
r
ap
h
'
s
d
en
s
ity
af
f
ec
ts
th
e
n
u
m
b
er
o
f
ch
r
o
m
atic
cl
ass
es
g
en
er
ated
b
y
th
e
alg
o
r
ith
m
s
.
As
th
e
g
r
a
p
h
d
en
s
ity
in
cr
ea
s
es
f
r
o
m
9
5
%
to
1
0
0
%,
th
e
n
u
m
b
er
o
f
c
o
lo
r
s
r
eq
u
ir
ed
to
c
o
lo
r
th
e
g
r
ap
h
r
is
es
s
ig
n
if
ican
tly
.
Ho
wev
er
,
f
o
r
g
r
a
p
h
d
en
s
ity
v
alu
es
ab
o
v
e
9
7
%,
b
o
th
al
g
o
r
ith
m
s
g
en
er
ate
id
en
tical
s
o
lu
tio
n
s
.
T
h
e
c
o
m
p
lex
ity
o
f
b
o
th
alg
o
r
ith
m
s
is
q
u
ad
r
atic,
p
r
im
ar
ily
d
ep
en
d
in
g
o
n
t
h
e
n
u
m
b
er
o
f
v
er
tices
in
th
e
g
r
a
p
h
a
n
d
,
to
a
less
er
ex
t
en
t,
o
n
its
d
e
n
s
ity
,
i.e
.
,
th
e
n
u
m
b
er
o
f
ed
g
es.
T
h
e
r
esu
lts
in
d
icate
th
at
with
a
s
ig
n
if
ican
t
in
cr
ea
s
e
in
g
r
ap
h
d
en
s
ity
,
i.e
.
,
a
s
u
b
s
tan
tial
in
cr
e
ase
in
th
e
n
u
m
b
er
o
f
ed
g
es,
th
e
ex
ec
u
tio
n
tim
e
o
f
b
o
th
alg
o
r
ith
m
s
ch
an
g
es in
s
ig
n
if
ican
tly
an
d
alm
o
s
t lin
ea
r
ly
,
with
o
n
ly
a
v
er
y
s
m
all
in
cr
em
en
t.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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t J E
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av
aila
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esp
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u
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VK,
u
p
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r
ea
s
o
n
ab
le
r
eq
u
est.
RE
F
E
R
E
NC
E
S
[
1
]
S
.
S
l
a
mi
n
,
N
.
O
.
A
d
i
w
i
j
a
y
a
,
M
.
A
.
H
a
sa
n
,
D
.
D
a
f
i
k
,
a
n
d
K
.
W
i
j
a
y
a
,
“
L
o
c
a
l
s
u
p
e
r
a
n
t
i
ma
g
i
c
t
o
t
a
l
l
a
b
e
l
i
n
g
f
o
r
v
e
r
t
e
x
c
o
l
o
r
i
n
g
o
f
g
r
a
p
h
s
,
”
S
y
m
m
e
t
ry
,
v
o
l
.
1
2
,
n
o
.
1
1
,
p
.
1
8
4
3
,
N
o
v
.
2
0
2
0
,
d
o
i
:
1
0
.
3
3
9
0
/
s
y
m1
2
1
1
1
8
4
3
.
[
2
]
T.
Em
d
e
n
-
W
e
i
n
e
r
t
,
S
.
H
o
u
g
a
r
d
y
,
a
n
d
B
.
K
r
e
u
t
e
r
,
“
U
n
i
q
u
e
l
y
c
o
l
o
u
r
a
b
l
e
g
r
a
p
h
s
a
n
d
t
h
e
h
a
r
d
n
e
ss
o
f
c
o
l
o
u
r
i
n
g
g
r
a
p
h
s
o
f
l
a
r
g
e
g
i
r
t
h
,
”
C
o
m
b
i
n
a
t
o
ri
c
s,
Pro
b
a
b
i
l
i
t
y
a
n
d
C
o
m
p
u
t
i
n
g
,
v
o
l
.
7
,
n
o
.
4
,
p
p
.
3
7
5
–
3
8
6
,
D
e
c
.
1
9
9
8
,
d
o
i
:
1
0
.
1
0
1
7
/
S
0
9
6
3
5
4
8
3
9
8
0
0
3
6
7
8
.
[
3
]
A
.
P
u
n
i
t
h
a
a
n
d
G
.
Ja
y
a
r
a
ma
n
,
“
C
o
m
p
u
t
a
t
i
o
n
o
f
t
o
t
a
l
c
h
r
o
m
a
t
i
c
n
u
mb
e
r
f
o
r
c
e
r
t
a
i
n
c
o
n
v
e
x
p
o
l
y
t
o
p
e
g
r
a
p
h
s
,
”
J
o
u
rn
a
l
o
f
A
p
p
l
i
e
d
Ma
t
h
e
m
a
t
i
c
s
a
n
d
I
n
f
o
rm
a
t
i
c
s
,
v
o
l
.
4
2
,
n
o
.
3
,
p
p
.
5
6
7
–
5
8
2
,
2
0
2
4
,
d
o
i
:
1
0
.
1
4
3
1
7
/
j
a
mi
.
2
0
2
4
.
5
6
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