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v
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m
An ana
ly
sis
bet
we
en t
he
Welsh
-
Pow
ell and DSa
t
ur al
g
o
rithms
for colo
ring
of sp
a
rse g
ra
phs
Ra
do
s
la
v
a
K
ra
lev
a
,
Velin K
ra
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T
o
ma
K
a
t
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rs
k
i
De
p
a
rtme
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t
o
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In
fo
rm
a
ti
c
s,
S
o
u
t
h
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Wes
t
Un
iv
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rsit
y
“
Ne
o
fit
Ril
s
k
i”,
Blag
o
e
v
g
ra
d
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Bu
l
g
a
ria
Art
icle
I
nfo
AB
S
T
RAC
T
A
r
ticle
his
to
r
y:
R
ec
eiv
ed
Au
g
2
2
,
2
0
2
4
R
ev
is
ed
Ma
r
2
0
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2
0
2
5
Acc
ep
ted
Ma
y
2
3
,
2
0
2
5
In
th
is
re
se
a
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h
a
n
a
n
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ly
sis
b
e
twe
e
n
th
e
Welsh
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P
o
we
ll
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n
d
DSatu
r
a
lg
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m
s
fo
r
t
h
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ra
p
h
v
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rte
x
c
o
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g
p
r
o
b
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wa
s
p
re
se
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ted
.
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o
th
a
lg
o
rit
h
m
s
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re
imp
lem
e
n
ted
a
n
d
a
n
a
l
y
z
e
d
a
s
we
ll
.
T
h
e
m
e
th
o
d
o
f
t
h
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e
x
p
e
rime
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t
wa
s
d
isc
u
ss
e
d
a
n
d
th
e
4
6
tes
t
g
ra
p
h
s,
w
h
ich
we
re
d
i
v
id
e
d
i
n
to
two
se
ts,
we
re
p
re
se
n
ted
.
Th
e
r
e
su
lt
s
sh
o
w
th
a
t
f
o
r
sp
a
rse
g
ra
p
h
s
with
a
sm
a
ll
e
r
n
u
m
b
e
r
o
f
v
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rti
c
e
s
a
n
d
e
d
g
e
s,
b
o
t
h
a
lg
o
rit
h
m
s
c
a
n
b
e
u
se
d
fo
r
so
lv
i
n
g
t
h
e
p
r
o
b
lem
.
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h
e
re
su
lt
s
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o
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th
a
t
in
5
0
%
o
f
th
e
c
a
se
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th
e
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P
o
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ll
a
l
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ri
th
m
f
o
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n
d
b
e
tt
e
r
so
l
u
ti
o
n
s
(2
3
i
n
to
tal)
.
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o
,
t
h
e
DSatu
r
a
lg
o
rit
h
m
fo
u
n
d
b
e
tt
e
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so
l
u
ti
o
n
s
in
o
n
ly
1
9
.
6
%
o
f
c
a
se
s
(9
in
t
o
t
a
l).
In
th
e
re
m
a
in
in
g
3
0
.
4
%
o
f
c
a
se
s,
b
o
t
h
a
lg
o
rit
h
m
s
fo
u
n
d
id
e
n
ti
c
a
l
so
lu
ti
o
n
s.
F
o
r
g
ra
p
h
s
with
a
larg
e
r
n
u
m
b
e
r
o
f
v
e
rti
c
e
s,
t
h
e
u
sa
g
e
o
f
th
e
Wel
sh
-
P
o
we
ll
a
lg
o
rit
h
m
is
re
c
o
m
m
e
n
d
e
d
a
s
it
fin
d
s
b
e
tt
e
r
so
l
u
ti
o
n
s.
T
h
e
e
x
e
c
u
ti
o
n
ti
m
e
o
f
th
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DSatu
r
a
lg
o
r
it
h
m
is
g
re
a
ter
th
a
n
th
e
e
x
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Wel
sh
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we
ll
a
lg
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rit
h
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,
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a
c
h
in
g
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p
t
o
a
m
i
n
u
te
f
o
r
g
ra
p
h
s
wit
h
a
lar
g
e
r
n
u
m
b
e
r
o
f
v
e
rti
c
e
s.
F
o
r
g
ra
p
h
s
wit
h
fe
we
r
v
e
rti
c
e
s
a
n
d
e
d
g
e
s,
th
e
e
x
e
c
u
ti
o
n
ti
m
e
s
o
f
b
o
t
h
a
l
g
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rit
h
m
s
a
re
sh
o
rter,
b
u
t
th
e
ti
m
e
is
st
il
l
g
re
a
ter
fo
r
t
h
e
DSatu
r
a
lg
o
rit
h
m
.
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
th
eo
r
y
Heu
r
is
tic
alg
o
r
ith
m
Ver
tex
co
lo
r
in
g
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
:
R
ad
o
s
lav
a
Kr
alev
a
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
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lag
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u
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E
m
ail: r
ad
y
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k
r
alev
a@
s
wu
.
b
g
1.
I
NT
RO
D
UCT
I
O
N
T
h
e
g
r
ap
h
-
t
y
p
e
s
tr
u
ct
u
r
e
a
n
d
th
e
alg
o
r
ith
m
s
ap
p
lied
to
th
is
s
tr
u
ctu
r
e
h
av
e
b
ee
n
wid
ely
an
d
th
o
r
o
u
g
h
ly
r
esear
ch
ed
in
t
h
e
a
r
ea
s
o
f
d
is
cr
ete
m
ath
em
atics
an
d
co
m
p
u
ter
s
cien
ce
in
th
e
las
t
f
ew
d
ec
ad
es
[
1
]
.
T
h
ese
s
tr
u
ctu
r
es
ar
e
u
s
ed
in
th
e
m
o
d
elin
g
o
f
m
a
n
y
p
r
o
ce
s
s
es
in
m
an
y
d
if
f
er
e
n
t
f
ield
s
o
f
s
cien
ce
an
d
p
r
ac
tice
[
2
]
.
T
h
is
m
ak
es
th
em
u
s
ef
u
l
an
d
o
f
te
n
u
s
ed
as
a
m
ea
n
s
o
f
an
aly
zin
g
ac
tiv
ities
,
ev
en
t
s
,
an
d
in
ter
ac
tio
n
s
b
etwe
en
d
if
f
er
en
t
o
b
jects
in
d
if
f
er
en
t
d
o
m
ain
s
o
f
th
e
r
ea
l
wo
r
ld
[
3
]
.
T
h
e
g
r
ap
h
-
ty
p
e
s
tr
u
ctu
r
es
ar
e
u
s
ed
to
p
r
esen
t
d
if
f
er
en
t
co
m
p
lex
p
r
o
b
lem
s
th
at
o
cc
u
r
o
f
ten
.
T
h
e
n
th
ey
ar
e
s
tu
d
ied
an
d
an
aly
ze
d
b
y
s
p
ec
ialized
s
o
f
twar
e
ap
p
licatio
n
s
th
at
ex
e
cu
te
d
if
f
er
e
n
t
alg
o
r
ith
m
s
o
n
t
h
ese
s
tr
u
ctu
r
es
[
4
]
.
T
h
er
ef
o
r
e
,
m
u
ch
r
esear
c
h
in
r
ec
en
t
y
ea
r
s
h
as
b
ee
n
aim
ed
at
im
p
r
o
v
in
g
an
d
o
p
tim
izin
g
v
ar
io
u
s
alg
o
r
ith
m
ic
m
eth
o
d
s
an
d
ap
p
r
o
ac
h
es
f
o
r
s
o
lv
in
g
d
if
f
e
r
en
t c
lass
es o
f
p
r
o
b
lem
s
m
o
d
eled
b
y
u
s
in
g
g
r
ap
h
s
tr
u
ctu
r
es
[
5
]
.
A
g
r
ap
h
-
t
y
p
e
s
tr
u
ctu
r
e
is
a
s
et
th
at
h
as
two
s
u
b
s
ets
–
a
s
et
o
f
v
er
tices
(
with
n
elem
en
ts
)
a
n
d
a
s
et
o
f
ed
g
es
(
with
m
elem
e
n
ts
)
.
A
v
er
tex
s
et
m
u
s
t
co
n
tain
at
least
o
n
e
elem
en
t
f
o
r
t
h
e
g
r
ap
h
its
elf
to
m
a
k
e
s
en
s
e.
Un
lik
e
th
e
s
et
o
f
v
er
tices,
th
e
s
et
o
f
ed
g
es m
ay
n
o
t h
av
e
an
y
elem
en
ts
.
T
h
is
ca
s
e
is
n
o
t ty
p
ical
o
f
a
g
r
ap
h
-
ty
p
e
s
tr
u
ctu
r
e,
alth
o
u
g
h
it
is
p
o
s
s
ib
le
s
in
ce
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g
es
r
ep
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c
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n
ec
tio
n
s
b
etwe
en
d
if
f
er
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t
p
air
s
o
f
v
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tices.
T
h
er
ef
o
r
e,
th
e
m
ain
id
ea
is
to
u
s
e
th
e
n
o
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-
lin
ea
r
d
ata
s
tr
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ctu
r
e,
wh
er
e
s
o
m
e
o
b
jects
(
v
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tices)
in
ter
ac
t
(
ar
e
m
u
tu
ally
d
e
p
en
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n
t)
in
a
ce
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t
ain
way
with
ea
ch
o
th
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,
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ac
co
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y
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th
ese
in
ter
ac
tio
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s
ar
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r
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ted
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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
.
4
,
Au
g
u
s
t
20
25
:
3
8
6
7
-
3875
3868
ed
g
es
[
6
]
.
I
f
ea
ch
e
d
g
e
is
ass
ig
n
ed
a
ce
r
tain
n
u
m
e
r
ical
v
alu
e,
th
en
th
e
g
r
ap
h
-
ty
p
e
s
tr
u
ctu
r
e
is
ca
lled
weig
h
ted
[
7
]
.
T
h
e
p
r
o
b
lem
o
f
co
lo
r
in
g
v
er
tices
in
a
g
r
ap
h
-
ty
p
e
s
tr
u
ctu
r
e
is
an
NP
-
h
ar
d
p
r
o
b
lem
[
8
]
.
Ho
wev
er
,
d
u
e
to
th
e
ex
t
r
em
e
ap
p
licab
ilit
y
an
d
im
p
o
r
tan
ce
o
f
t
h
is
p
r
o
b
l
em
,
it
co
n
tin
u
es
to
b
e
ac
tiv
ely
r
esear
ch
ed
[
9
]
.
T
h
er
ef
o
r
e,
th
er
e
ar
e
also
m
an
y
d
if
f
e
r
en
t
v
a
r
ian
ts
(
as
d
ef
in
i
tio
n
s
)
o
f
t
h
is
p
r
o
b
lem
.
Fo
r
e
x
am
p
l
e,
th
e
v
er
tex
co
lo
r
in
g
with
co
m
m
u
n
icatio
n
co
n
s
tr
ain
ts
in
s
y
n
c
h
r
o
n
o
u
s
b
r
o
ad
ca
s
t
n
etwo
r
k
s
[
1
0
]
,
t
h
e
r
ec
o
n
f
ig
u
r
atio
n
g
r
ap
h
f
o
r
v
er
tex
c
o
lo
r
in
g
s
o
f
wea
k
ly
ch
o
r
d
al
g
r
ap
h
s
[
1
1
]
,
th
e
f
a
cial
u
n
iq
u
e
-
m
ax
im
u
m
co
lo
r
in
g
s
o
f
p
lan
e
g
r
ap
h
s
with
r
estrictio
n
o
n
b
ig
v
er
tic
es
[
1
2
]
,
an
d
m
an
y
o
th
e
r
s
.
Sep
ar
ate
way
s
to
s
o
lv
e
th
is
p
r
o
b
lem
u
s
e
d
if
f
er
e
n
t
alg
o
r
ith
m
s
[
1
3
]
,
d
if
f
e
r
en
t
tech
n
iq
u
es
[
1
4
]
,
an
d
d
if
f
er
e
n
t
ap
p
r
o
ac
h
es
[
1
5
]
.
T
h
ese
m
eth
o
d
s
f
o
r
s
o
lv
in
g
th
is
p
r
o
b
lem
h
a
v
e
also
b
ee
n
u
s
ed
in
s
o
lv
in
g
o
th
er
p
r
o
b
lem
s
m
o
d
eled
b
y
g
r
ap
h
-
ty
p
e
s
tr
u
c
tu
r
es
[
1
6
]
.
A
m
o
r
e
co
m
p
r
eh
e
n
s
iv
e
p
r
esen
tatio
n
o
f
th
e
g
r
ap
h
v
e
r
tex
c
o
lo
r
in
g
p
r
o
b
lem
is
also
p
r
esen
ted
in
o
t
h
er
s
cien
tific
wo
r
k
s
[
1
7
]
–
[
2
0
]
.
An
y
g
r
ap
h
th
at
is
n
o
t
ac
y
clic,
n
o
t
co
m
p
lete,
a
n
d
d
o
es
n
o
t
h
a
v
e
a
cy
cle
o
f
o
d
d
len
g
th
ca
n
b
e
co
lo
r
ed
with
s
ev
er
al
co
lo
r
s
th
at
ar
e
e
q
u
a
l
to
o
r
less
th
an
th
e
lar
g
e
s
t
d
eg
r
ee
o
f
a
v
er
tex
in
t
h
at
g
r
ap
h
.
Pro
o
f
o
f
th
is
s
tatem
en
t
is
p
r
esen
ted
in
[
2
1
]
.
I
f
χ(
G)
d
en
o
tes
th
e
ch
r
o
m
atic
n
u
m
b
e
r
o
f
a
g
i
v
en
g
r
ap
h
,
an
d
∆(
G)
d
en
o
tes
th
e
lar
g
est
d
eg
r
ee
o
f
a
v
e
r
tex
i
n
th
is
g
r
ap
h
,
th
e
n
th
e
ab
o
v
e
s
tatem
en
t
ca
n
b
e
p
r
esen
ted
i
n
th
e
f
o
llo
win
g
way
:
χ(
G
)
≤
∆(
G)
.
Mo
r
eo
v
er
,
if
f
o
r
all
v
er
tices
in
a
g
i
v
en
g
r
ap
h
s
tr
u
ct
u
r
e,
it
is
tr
u
e
th
at
th
e
n
u
m
b
e
r
o
f
ed
g
es
in
cid
en
t
to
th
ese
v
er
tices
is
g
r
ea
ter
th
an
2
,
i.e
.
ev
er
y
v
e
r
tex
in
th
is
g
r
ap
h
s
tr
u
ctu
r
e
h
as
d
eg
r
ee
g
r
ea
ter
th
an
2
,
t
h
en
t
h
e
n
u
m
b
er
o
f
co
lo
r
s
r
e
q
u
ir
ed
to
co
lo
r
th
e
g
iv
en
g
r
ap
h
will
b
e
o
n
e
g
r
ea
ter
th
an
th
e
lar
g
est
d
eg
r
ee
o
f
a
v
e
r
tex
in
th
e
s
am
e
g
r
ap
h
,
an
d
o
n
ly
if
in
th
at
g
r
ap
h
th
er
e
is
a
cliq
u
e
th
at
co
n
tain
s
e
x
ac
tly
∆(
G
)
+1
v
er
tices
[
2
1
]
.
Oth
e
r
r
esu
lts
r
elate
d
to
th
e
p
r
o
b
lem
o
f
f
in
d
in
g
th
e
ch
r
o
m
atic
n
u
m
b
er
o
f
a
g
iv
en
g
r
ap
h
,
as
w
ell
as
f
in
d
in
g
lo
wer
b
o
u
n
d
s
f
o
r
th
is
n
u
m
b
er
,
ar
e
p
r
esen
ted
in
[
2
2
]
.
T
h
er
e
a
r
e
m
a
n
y
o
th
er
alg
o
r
ith
m
s
(
an
d
ap
p
r
o
ac
h
es)
f
o
r
f
i
n
d
in
g
th
e
ch
r
o
m
atic
n
u
m
b
e
r
s
o
f
g
r
ap
h
s
,
wh
ich
ar
e
p
r
esen
ted
in
d
et
ail
in
v
ar
io
u
s
s
cien
tific
p
u
b
licatio
n
s
[
2
3
]
–
[
2
5
]
.
I
n
th
is
p
ap
er
,
two
ap
p
r
o
x
im
at
e
(
h
eu
r
is
tic)
alg
o
r
ith
m
s
f
o
r
s
o
lv
in
g
th
e
g
r
ap
h
v
er
tex
co
l
o
r
in
g
p
r
o
b
lem
will
b
e
an
aly
ze
d
,
r
esp
ec
tiv
el
y
–
DSatu
r
[
2
6
]
a
n
d
W
elsh
-
Po
well
[
2
7
]
.
T
h
ese
alg
o
r
ith
m
s
a
r
e
ap
p
r
o
x
im
ate
an
d
d
o
n
o
t
alwa
y
s
f
in
d
an
o
p
tim
al
s
o
lu
tio
n
in
ter
m
s
o
f
th
e
n
u
m
b
er
o
f
g
e
n
er
atio
n
s
o
f
th
e
ch
r
o
m
atic
class
e
s
.
T
h
is
m
ea
n
s
th
at
th
e
v
er
tices o
f
a
g
i
v
en
g
r
a
p
h
d
iv
i
d
e
in
to
in
d
e
p
en
d
en
t su
b
s
e
ts
o
f
v
er
tices.
T
h
is
s
tu
d
y
aim
s
to
d
ec
id
e
wh
ich
o
f
th
e
two
alg
o
r
ith
m
s
will
f
in
d
b
etter
s
o
lu
tio
n
s
f
o
r
g
r
ap
h
s
with
a
p
r
e
d
ef
in
ed
n
u
m
b
er
o
f
v
e
r
tices
an
d
ed
g
es.
2.
RE
S
E
ARCH
M
E
T
H
O
D
T
h
is
s
ec
tio
n
p
r
esen
ts
im
p
lem
en
tatio
n
s
o
f
th
e
W
elsh
-
Po
well
an
d
DSatu
r
alg
o
r
ith
m
s
.
B
o
th
alg
o
r
ith
m
s
ar
e
ap
p
r
o
x
im
ate
an
d
u
s
ed
to
s
o
lv
e
th
e
g
r
ap
h
v
e
r
tex
c
o
lo
r
in
g
p
r
o
b
lem
.
A
p
ar
t
o
f
g
lo
b
al
p
ar
am
eter
s
an
d
ar
r
a
y
s
m
u
s
t b
e
p
r
e
-
d
ec
lar
ed
f
o
r
b
o
th
alg
o
r
ith
m
s
.
T
h
e
y
ar
e
p
r
esen
te
d
in
Fig
u
r
e
1
.
01
var
02
│
CountOfVertices: Integer;
03
│
MinimalColorCount: Integer;
04
│
VectorOfColors:
array
of
TColor;
05
│
AdjacencyStucture:
array
of
array
of
Integer;
06
│
VertexList:
array
of
Integer; EdgeList:
array
of
Integer;
Fig
u
r
e
1
.
Pre
-
d
ec
lar
ed
g
lo
b
al
p
ar
am
eter
s
an
d
a
r
r
ay
s
T
h
e
C
o
u
n
tOfVe
r
tices
p
ar
am
eter
(
d
ec
lar
ed
o
n
lin
e
2
)
co
n
tai
n
s
th
e
n
u
m
b
e
r
o
f
v
er
tices
in
th
e
g
r
ap
h
d
ata
s
tr
u
ctu
r
e.
T
h
e
Min
ima
lC
o
lo
r
C
o
u
n
t
p
ar
am
eter
(
d
ec
lar
e
d
o
n
lin
e
3
)
is
an
ad
d
itio
n
al
p
ar
am
eter
u
s
ed
b
y
th
e
alg
o
r
ith
m
s
in
th
e
d
ec
is
io
n
p
r
o
ce
s
s
.
T
h
e
V
ec
to
r
OfC
o
lo
r
s
v
ec
to
r
(
o
f
ty
p
e
TC
o
lo
r
)
,
d
e
clar
ed
o
n
lin
e
3
,
is
u
s
ed
b
y
alg
o
r
ith
m
s
to
s
to
r
e
th
e
ar
r
ay
o
f
ass
o
r
ted
co
lo
r
s
.
E
ac
h
it
em
in
th
is
v
ec
to
r
s
to
r
es
a
co
lo
r
with
wh
ich
th
e
g
iv
en
v
er
tex
o
f
t
h
e
g
r
ap
h
d
at
a
s
tr
u
ctu
r
e
is
co
l
o
r
ed
.
E
ac
h
s
tr
u
ctu
r
e
o
f
th
e
g
r
a
p
h
ty
p
e
is
r
ep
r
esen
ted
b
y
a
n
ad
jace
n
cy
m
atr
ix
s
tr
u
ctu
r
e
(
d
ec
lar
ed
o
n
lin
e
5
)
,
a
v
er
te
x
lis
t
v
ec
to
r
,
an
d
a
n
ed
g
e
lis
t
v
ec
t
o
r
(
d
ec
lar
e
d
o
n
lin
e
6
)
.
E
ac
h
item
(
s
,
f)
o
f
th
e
A
d
ja
ce
n
cy
S
tr
u
ctu
r
e
in
d
icate
s
wh
et
h
er
th
e
v
e
r
ti
ce
s
with
in
d
ices
s
(
s
tar
t)
an
d
f
(
f
in
al
)
ar
e
ad
jace
n
t o
r
n
o
t.
T
h
e
Wels
h
P
o
w
ellA
lg
o
r
ith
m
p
r
o
ce
d
u
r
e
im
p
lem
en
ts
th
e
f
i
r
s
t
h
eu
r
is
tic
alg
o
r
ith
m
f
o
r
c
o
lo
r
in
g
th
e
v
er
tices
o
f
a
g
r
a
p
h
.
T
h
e
s
o
u
r
ce
co
d
e
o
f
th
is
p
r
o
ce
d
u
r
e
i
s
s
h
o
wn
in
Fig
u
r
e
2
.
I
t
u
s
es
ad
d
itio
n
al
(
lo
ca
l)
p
ar
am
eter
s
-
C
o
lo
r
C
o
u
n
t
,
C
o
lo
r
ed
V
erti
ce
s
C
o
u
n
t
,
A
cc
ep
tDec
is
io
n
,
I
tera
tio
n
,
an
d
V
ertex
.
T
h
e
I
tera
tio
n
an
d
V
ertex
p
ar
am
eter
s
ar
e
u
s
ed
f
o
r
iter
atio
n
v
ar
iab
les
o
f
th
e
t
wo
fo
r
lo
o
p
s
e
x
ec
u
ted
o
n
li
n
es
9
an
d
1
4
.
T
h
e
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
A
n
a
n
a
lysi
s
b
etw
ee
n
th
e
Wels
h
-
P
o
w
ell
a
n
d
D
S
a
tu
r
a
lg
o
r
ith
ms fo
r
…
(
R
a
d
o
s
la
va
K
r
a
leva
)
3869
C
o
lo
r
C
o
u
n
t
p
ar
am
eter
s
to
r
es
t
h
e
n
u
m
b
er
o
f
o
n
e
o
f
th
e
co
lo
r
s
u
s
ed
s
o
f
a
r
.
T
h
e
A
cc
e
p
tDec
is
io
n
p
ar
am
eter
(
o
f
B
o
o
lea
n
t
y
p
e)
s
h
o
ws wh
eth
er
th
e
g
iv
en
v
er
tex
ca
n
b
e
c
o
lo
r
e
d
with
o
n
e
o
f
th
e
a
v
ailab
le
co
l
o
r
s
o
r
n
o
t.
T
h
e
p
ar
am
eter
C
o
lo
r
ed
V
erti
c
esC
o
u
n
t
s
to
r
es
th
e
cu
r
r
en
t
n
u
m
b
er
o
f
c
o
lo
r
e
d
v
er
tices
in
th
e
g
r
ap
h
.
I
n
th
e
W
elsh
Po
well
alg
o
r
ith
m
,
t
h
e
p
ar
a
m
eter
s
Min
ima
lC
o
lo
r
C
o
u
n
t
,
C
o
lo
r
C
o
u
n
t
,
C
o
lo
r
ed
V
erti
ce
s
C
o
u
n
t
ar
e
s
et
to
0
(
lin
es 3
-
5
)
.
Acc
o
r
d
in
g
ly
,
t
h
e
A
cc
ep
tDec
is
io
n
v
ar
iab
le
is
s
et
to
F
a
ls
e
(
lin
e
6
)
.
01
procedure
WelshPowellAlgorithm;
02
begin
03 │
MinimalColorCount := 0;
04 │
var
ColorCount := 0;
05 │
var
ColoredVerticesCount := 0;
06 │
var
AcceptDecision := False;
07 │
repeat
08 │ │
ColorCount := ColorCount + 1;
09 │ │
for var
Iteration := 1
to
CountOfVertices
do
10 │ │
begin
11 │ │ │
if
(VectorOfColors[Iteration] = 0)
then
12 │ │ │
begin
13 │ │ │ │
AcceptDecision := True;
14 │ │ │ │
for var
Vertex := 1
to
CountOfVertices
do
15 │ │ │ │
begin
16 │ │ │ │ │
if
((AdjacencyStucture[
Iteration][Vertex] > 0)
and
17 │ │ │ │ │
(VectorOfColors[Vertex] = ColorCount))
then
18 │ │ │ │ │
begin
19 │ │ │ │ │ │
AcceptDecision := False;
20 │ │ │ │ │ │
Break;
21 │ │ │ │ │
end
;
22 │ │ │ │
end
;
23 │ │ │ │
if
(AcceptDecision = True)
then
24 │ │ │
│
begin
25 │ │ │ │ │
VectorOfColors[Iteration] := ColorCount;
26 │ │ │ │ │
if
(MinimalColorCount < ColorCount)
then
27 │ │ │ │ │
MinimalColorCount := ColorCount;
28 │ │ │ │ │
ColoredVerticesCount := ColoredVerticesCount + 1;
29 │ │ │ │
end
;
30 │ │ │
end
;
31 │ │
end
;
32 │
until
(ColoredVerticesCount = CountOfVertices);
33
end
;
Fig
u
r
e
2
.
So
u
r
ce
co
d
e
o
f
th
e
W
elsh
Po
well
Alg
o
r
ith
m
p
r
o
ce
d
u
r
e
T
h
e
alg
o
r
ith
m
is
ex
ec
u
ted
u
n
t
il
all
v
er
tices
in
th
e
g
r
ap
h
ar
e
co
lo
r
ed
(
lin
e
3
2
-
th
e
“u
n
til”
co
n
d
itio
n
f
o
r
th
e
e
n
d
o
f
th
e
r
e
p
ea
t
lo
o
p
)
.
At
th
e
b
e
g
in
n
in
g
o
f
ea
ch
iter
atio
n
o
f
t
h
e
r
ep
ea
t
lo
o
p
,
a
n
ew
co
lo
r
in
d
ex
(
n
u
m
b
e
r
)
is
s
elec
ted
(
lin
e
8
)
.
At
th
is
s
tep
o
f
th
e
alg
o
r
ith
m
'
s
ex
ec
u
tio
n
,
a
t
r
av
er
s
al
o
f
t
h
e
v
er
tices
o
f
t
h
e
g
r
ap
h
b
eg
in
s
(
lin
e
9
)
.
I
f
th
e
c
u
r
r
e
n
tly
an
aly
ze
d
v
er
tex
is
n
o
t
co
l
o
r
ed
(
s
ee
lin
e
1
1
)
a
n
d
th
at
v
e
r
t
ex
h
as
n
o
ad
jace
n
t
v
er
tices
th
at
ar
e
co
l
o
r
ed
with
th
at
co
lo
r
,
th
en
t
h
e
cu
r
r
en
t
v
e
r
tex
is
co
lo
r
e
d
with
th
e
cu
r
r
e
n
t
co
lo
r
(
lin
e
2
5
)
.
I
f
am
o
n
g
th
e
ad
jace
n
cy
v
er
tices
o
f
th
e
cu
r
r
en
t
v
er
tex
is
n
o
t
co
lo
r
ed
with
th
e
cu
r
r
en
t
co
lo
r
,
th
en
th
e
lo
g
ical
v
ar
iab
le
A
cc
ep
tDec
is
io
n
will
h
av
e
th
e
v
alu
e
Tr
u
e
.
On
l
y
in
th
is
ca
s
e
will
th
e
cu
r
r
en
t
v
e
r
tex
b
e
co
lo
r
ed
with
th
e
cu
r
r
e
n
t
co
lo
r
.
T
h
e
s
o
u
r
ce
co
d
e
o
f
lin
e
2
6
ch
ec
k
s
wh
et
h
er
th
e
p
ar
a
m
eter
C
o
lo
r
C
o
u
n
t
is
g
r
ea
ter
th
an
th
e
v
alu
e
o
f
th
e
p
ar
a
m
eter
Min
ima
lC
o
lo
r
C
o
u
n
t
.
I
f
th
is
is
th
e
ca
s
e,
th
en
th
e
v
alu
e
o
f
th
e
C
o
lo
r
C
o
u
n
t
p
ar
am
eter
is
ass
ig
n
ed
as
th
e
v
alu
e
o
f
th
e
Min
ima
lC
o
lo
r
C
o
u
nt
p
ar
am
eter
.
As
an
o
th
er
v
er
te
x
h
as
b
ee
n
s
u
cc
ess
f
u
lly
co
lo
r
ed
,
th
is
is
r
ec
o
r
d
ed
b
y
u
p
d
ati
n
g
t
h
e
v
alu
e
o
f
t
h
e
C
o
lo
r
ed
V
erti
ce
s
C
o
u
n
t
p
ar
am
eter
,
wh
ich
is
in
cr
em
en
ted
b
y
1
.
T
h
e
co
m
p
u
tatio
n
al
co
m
p
lex
it
y
o
f
th
is
alg
o
r
ith
m
is
q
u
ad
r
at
ic
an
d
d
ep
en
d
s
o
n
th
e
n
u
m
b
e
r
o
f
v
er
tices
o
f
th
e
g
r
ap
h
(
th
e
C
o
u
n
tOfVe
r
tices
p
ar
am
eter
)
.
A
s
im
ilar
im
p
lem
e
n
tatio
n
o
f
th
is
alg
o
r
ith
m
is
p
r
ese
n
ted
in
[
2
0
]
.
T
h
e
DS
a
tu
r
A
lg
o
r
ith
m
p
r
o
ce
d
u
r
e
im
p
lem
en
ts
th
e
s
ec
o
n
d
h
eu
r
is
tic
alg
o
r
ith
m
f
o
r
co
lo
r
i
n
g
th
e
v
er
tices
o
f
a
g
r
ap
h
.
T
h
e
s
o
u
r
ce
co
d
e
o
f
th
is
p
r
o
ce
d
u
r
e
is
s
h
o
wn
in
Fig
u
r
e
3
.
I
t
also
u
s
es
ad
d
itio
n
al
p
ar
am
eter
s
–
C
o
lo
r
C
o
u
n
t
,
C
o
lo
r
ed
V
erti
ce
s
C
o
u
n
t
,
A
cc
ep
tDec
is
io
n
,
an
d
V
ertex
.
T
h
e
V
ertex
p
ar
am
eter
s
to
r
es
th
e
n
u
m
b
e
r
(
in
d
ex
)
o
f
th
e
n
ex
t
v
er
tex
th
a
t
will
b
e
co
lo
r
ed
.
T
h
e
C
o
lo
r
C
o
u
n
t
p
ar
am
eter
s
to
r
es
th
e
n
u
m
b
er
o
f
o
n
e
o
f
th
e
co
lo
r
s
u
s
ed
s
o
f
ar
.
T
h
e
p
ar
am
eter
C
o
lo
r
ed
V
erti
c
esC
o
u
n
t
s
to
r
es
th
e
cu
r
r
en
t
n
u
m
b
er
o
f
c
o
lo
r
e
d
v
er
tices
in
th
e
g
r
ap
h
.
I
n
th
e
DSatu
r
alg
o
r
ith
m
,
th
e
p
ar
am
eter
s
Min
ima
lC
o
lo
r
C
o
u
n
t
,
V
ertex
,
C
o
lo
r
C
o
u
n
t
,
an
d
C
o
l
o
r
ed
V
erti
ce
s
C
o
u
n
t
ar
e
s
et
to
0
(
lin
e
s
3
-
6
)
.
T
h
e
al
g
o
r
ith
m
is
e
x
ec
u
ted
u
n
til
all
v
er
tices
in
th
e
g
r
a
p
h
a
r
e
co
l
o
r
e
d
(
lin
e
7
-
th
e
wh
ile
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
.
4
,
Au
g
u
s
t
20
25
:
3
8
6
7
-
3875
3870
co
n
d
itio
n
f
o
r
th
e
b
eg
in
n
in
g
o
f
th
e
wh
ile
lo
o
p
)
.
At
th
e
b
eg
i
n
n
in
g
o
f
ea
ch
iter
atio
n
o
f
th
e
wh
ile
lo
o
p
,
th
e
n
ex
t
in
d
ex
(
n
u
m
b
e
r
)
o
f
th
e
v
er
tex
i
s
s
elec
ted
(
lin
e
9
)
.
T
h
e
F
o
u
n
d
N
ex
tVe
r
tex
f
u
n
ctio
n
s
elec
ts
th
e
n
ex
t
v
er
tex
to
b
e
co
lo
r
ed
.
T
h
is
v
er
tex
m
u
s
t
f
u
l
f
ill
o
n
e
o
f
th
e
f
o
llo
win
g
f
o
u
r
cr
iter
ia
(
in
th
e
o
r
d
er
in
wh
ic
h
th
ey
ar
e
lis
ted
)
.
I
t
m
u
s
t
h
av
e
th
e
h
i
g
h
est
v
alu
e
o
f
th
e
d
e
g
r
e
e
o
f
s
atu
r
atio
n
ch
ar
ac
ter
is
tic.
I
n
th
e
ca
s
e
th
at
th
er
e
is
m
o
r
e
th
an
o
n
e
v
er
te
x
with
s
u
ch
a
v
alu
e,
a
s
ec
o
n
d
s
elec
tio
n
cr
iter
io
n
is
u
s
ed
.
A
cc
o
r
d
in
g
to
it,
th
e
v
er
tex
m
u
s
t
h
av
e
th
e
lar
g
est
r
esid
u
al
d
eg
r
ee
in
th
e
g
r
ap
h
in
d
u
ce
d
b
y
th
e
cu
r
r
en
t
o
n
e
a
f
ter
r
em
o
v
in
g
t
h
e
co
lo
r
ed
v
er
t
ices.
I
f
th
er
e
ar
e
m
o
r
e
v
er
tice
s
,
a
th
ir
d
s
ele
ctio
n
cr
iter
io
n
is
u
s
ed
.
Acc
o
r
d
in
g
t
o
it,
th
at
v
er
tex
is
s
elec
ted
,
w
h
ich
ca
n
b
e
co
lo
r
e
d
with
a
co
lo
r
with
a
s
m
aller
n
u
m
b
er
.
I
n
th
e
last
ca
s
e,
th
e
v
er
tex
with
th
e
s
m
allest
n
u
m
b
e
r
is
s
elec
ted
am
o
n
g
all
v
er
tices
h
av
in
g
th
e
s
am
e
v
alu
es
o
f
th
e
ab
o
v
e
th
r
ee
cr
ite
r
ia.
At
th
is
s
tep
,
th
e
ch
o
s
en
v
e
r
tex
(
in
d
e
x
ed
b
y
th
e
V
ertex
p
a
r
am
eter
)
is
co
lo
r
ed
with
th
e
cu
r
r
en
t
c
o
lo
r
.
T
h
e
v
alu
e
o
f
th
e
C
o
lo
r
ed
V
erti
ce
s
C
o
u
n
t
p
ar
am
eter
is
in
cr
em
en
te
d
b
y
o
n
e
af
ter
th
e
s
u
cc
ess
f
u
l c
o
lo
r
in
g
o
f
th
e
v
er
t
ex
(
lin
e
1
3
)
.
01
procedure
DSaturAlgo
rithm;
02
begin
03
│
MinimalColorCount := 0;
04
│
var
Vertex := 0;
05
│
var
ColorCount := 0;
06
│
var
ColoredVerticesCount := 0;
07
│
while not
(ColoredVerticesCount = CountOfVertices)
do
08
│
begin
09
│
│
Vertex := FoundNextVertex;
10
│
│
if
(Vertex > 0)
then
11
│
│
begin
12
│
│
│
VectorOfColors[Vertex] := ColorCount;
13
│
│
│
ColoredVerticesCount := ColoredVerticesCount + 1;
14
│
│
│
UpdateParameters();
15
│
│
│
UpdateColorCount(ColorCount);
16 │ │ │
if
(MinimalColorCount <
ColorCount)
then
17 │ │ │
MinimalColorCount := ColorCount;
18
│
│
end
;
19
│
end
;
20
end
;
Fig
u
r
e
3
.
So
u
r
ce
co
d
e
o
f
th
e
DSatu
r
Alg
o
r
ith
m
p
r
o
ce
d
u
r
e
T
h
e
Up
d
a
teP
a
r
a
mete
r
s
p
r
o
ce
d
u
r
e
u
p
d
ates
th
e
v
alu
es
o
f
th
e
d
eg
r
ee
o
f
s
atu
r
atio
n
an
d
r
esid
u
al
d
eg
r
ee
p
ar
am
eter
s
(
lin
e
1
4
)
.
T
h
e
Up
d
a
teC
o
lo
r
C
o
u
n
t
p
r
o
ce
d
u
r
e,
w
h
ich
r
ec
eiv
es
an
in
p
u
t
p
ar
am
eter
–
C
o
lo
r
C
o
u
n
t
(
p
ass
ed
b
y
ad
d
r
ess
)
,
in
cr
em
e
n
ts
th
e
n
u
m
b
er
o
f
av
aila
b
le
co
lo
r
s
if
n
ec
ess
ar
y
(
lin
e
1
4
)
.
T
h
e
s
o
u
r
ce
co
d
e
o
f
lin
e
1
6
ch
ec
k
s
wh
eth
er
t
h
e
p
ar
am
e
ter
C
o
lo
r
C
o
u
n
t
is
g
r
ea
ter
th
a
n
th
e
v
alu
e
o
f
th
e
p
a
r
am
eter
Mi
n
ima
lC
o
lo
r
C
o
u
n
t
.
I
n
th
is
ca
s
e,
th
e
v
alu
e
o
f
th
e
C
o
lo
r
C
o
u
n
t
p
ar
am
eter
is
ass
ig
n
ed
as
th
e
v
alu
e
o
f
th
e
Min
ima
lC
o
lo
r
C
o
u
n
t
p
ar
am
eter
.
T
h
e
co
m
p
u
tatio
n
al
co
m
p
le
x
ity
o
f
th
is
alg
o
r
ith
m
is
also
q
u
ad
r
atic
an
d
d
e
p
en
d
s
o
n
th
e
n
u
m
b
er
o
f
v
er
tices o
f
th
e
g
r
ap
h
(
C
o
u
n
tOf
V
erti
ce
s
p
ar
am
eter
)
.
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
er
im
en
t
ar
e
s
h
o
wn
an
d
d
is
cu
s
s
ed
.
Fo
r
th
is
s
tu
d
y
,
4
6
s
p
ar
s
e
g
r
a
p
h
s
,
r
esp
ec
tiv
ely
with
8
0
0
,
1
0
0
0
,
1
2
0
0
,
.
.
.
,
9
6
0
0
,
an
d
9
8
0
0
v
er
tices
wer
e
cr
ea
ted
.
E
ac
h
g
r
ap
h
h
as
a
d
en
s
ity
o
f
3
%.
B
ased
o
n
th
is
ch
ar
ac
ter
is
tic
o
f
th
e
g
r
ap
h
ex
ac
tly
3
%
o
f
am
o
n
g
all
v
er
tices,
ar
e
r
an
d
o
m
ly
s
elec
ted
.
T
h
ese
g
r
a
p
h
s
ar
e
p
r
esen
ted
in
T
ab
le
1
.
Mo
r
e
i
n
f
o
r
m
atio
n
,
w
h
ich
in
cl
u
d
es
t
h
e
r
esu
lts
o
f
th
e
ex
ec
u
tio
n
o
f
th
e
two
h
e
u
r
is
tic
alg
o
r
ith
m
s
,
in
ter
m
s
o
f
th
eir
ex
ec
u
tio
n
tim
e
a
n
d
th
e
q
u
alit
y
o
f
th
e
g
en
e
r
ated
s
o
lu
tio
n
s
i
s
s
h
o
wn
in
T
ab
le
2
.
T
h
e
ex
p
e
r
im
en
tal
co
n
d
itio
n
s
ar
e:
6
4
-
b
it
W
in
d
o
ws
1
1
a
n
d
h
ar
d
war
e
co
n
f
ig
u
r
a
tio
n
:
Pro
ce
s
s
o
r
:
I
n
tel
(
R
)
C
o
r
e
(
T
M)
i5
-
1
2
4
5
0
H
at
4
.
4
0
GHz
;
R
AM
: 1
6
GB
,
SS
D
1
0
0
0
GB
.
I
n
T
ab
le
s
1
an
d
2
,
t
h
e
"
Gra
p
h
I
D
"
co
l
u
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
"
Gra
p
h
file
n
a
me
"
co
lu
m
n
s
h
o
ws
th
e
n
am
e
o
f
th
e
f
ile
in
wh
ich
th
e
in
f
o
r
m
atio
n
f
o
r
th
e
co
r
r
esp
o
n
d
in
g
t
est
g
r
ap
h
is
s
to
r
ed
.
T
h
e
"
V
ertex
co
u
n
t
"
co
lu
m
n
s
h
o
ws
th
e
n
u
m
b
er
o
f
v
er
tices
o
f
a
g
r
ap
h
.
T
h
e
"
Ma
x
ed
g
e
c
o
u
n
t
"
co
l
u
m
n
s
h
o
ws
th
e
1
0
0
%
d
en
s
ity
o
f
a
g
r
ap
h
i
n
ter
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5
9
8
8
0
42
3
7
6
5
43
4
5
8
4
4
15
3
6
0
0
1
9
4
3
4
6
25
5
1
6
24
6
1
8
7
38
8
2
0
0
1
0
0
8
4
7
8
43
3
9
3
7
44
4
8
3
1
2
16
3
8
0
0
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6
5
4
4
26
5
9
4
24
7
1
5
6
39
8
4
0
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0
5
8
2
7
4
43
4
3
5
9
44
5
1
2
5
0
17
4
0
0
0
2
3
9
9
4
0
26
6
4
0
25
8
1
2
5
40
8
6
0
0
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1
0
9
2
7
2
45
4
6
0
9
45
5
4
3
1
3
18
4
2
0
0
2
6
4
5
3
8
27
7
5
0
27
9
0
3
2
41
8
8
0
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1
4
6
8
44
4
9
8
5
45
5
8
9
6
9
19
4
4
0
0
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9
0
3
3
4
28
8
1
3
27
1
0
0
0
0
42
9
0
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0
1
2
1
4
8
6
6
47
5
1
8
8
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6
2
3
2
8
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4
6
0
0
3
1
7
3
3
2
29
9
2
2
28
1
1
0
3
1
43
9
2
0
0
1
2
6
9
4
6
2
47
5
5
1
6
47
6
6
1
5
7
21
4
8
0
0
3
4
5
5
2
8
30
1
0
0
0
30
1
2
2
6
5
44
9
4
0
0
1
3
2
5
2
6
0
48
6
0
9
4
47
6
9
9
3
8
22
5
0
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0
3
7
4
9
2
6
30
1
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7
2
30
1
3
5
7
8
45
9
6
0
0
1
3
8
2
2
5
6
48
6
1
2
5
48
7
6
6
1
0
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5
2
0
0
4
0
5
5
2
2
32
1
2
8
1
31
1
5
0
6
3
46
9
8
0
0
1
4
4
0
4
5
4
49
6
5
4
7
50
7
9
1
7
2
T
ab
le
2
a
n
d
th
e
ch
ar
ts
in
Fig
u
r
e
s
4
,
5
,
an
d
6
s
h
o
w
th
e
r
esu
lts
o
f
th
e
two
h
eu
r
is
tic
alg
o
r
ith
m
s
f
o
r
th
e
g
r
ap
h
s
p
r
esen
ted
in
T
a
b
le
1
.
T
h
e
r
esu
lts
in
th
e
co
lu
m
n
s
"
C
o
lo
r
"
an
d
"
Time
(
ms)
"
(
f
o
r
b
o
th
W
elsh
-
Po
well
an
d
DSatu
r
alg
o
r
ith
m
s
)
s
h
o
w
t
h
e
n
u
m
b
er
o
f
co
l
o
r
s
n
ee
d
ed
(
m
o
r
e
p
r
ec
is
ely
,
th
e
n
u
m
b
er
o
f
ch
r
o
m
atic
class
es)
to
co
lo
r
th
e
co
r
r
esp
o
n
d
in
g
g
r
ap
h
,
as
well
as
th
e
tim
e
to
f
in
d
an
ac
ce
p
tab
le
s
o
lu
tio
n
.
T
h
e
r
esu
lts
s
h
o
w
th
at
in
th
e
f
ir
s
t
h
alf
o
f
th
e
s
et
o
f
g
r
a
p
h
s
(
1
–
2
3
)
in
o
n
l
y
f
iv
e
o
u
t
o
f
a
to
tal
o
f
2
3
ca
s
es
(
f
o
r
g
r
ap
h
s
7
,
1
2
,
1
8
,
2
1
,
a
n
d
2
2
)
th
e
W
elsh
-
Po
well
alg
o
r
ith
m
f
o
u
n
d
t
h
e
s
am
e
s
o
lu
tio
n
s
as th
e
DSatu
r
alg
o
r
ith
m
.
T
ab
le
2
an
d
th
e
ch
ar
ts
in
Fig
u
r
e
s
4
an
d
5
s
h
o
w
th
at
f
o
r
th
e
co
m
p
lete
s
et
o
f
g
r
ap
h
s
in
o
n
l
y
1
4
ca
s
es,
b
o
th
alg
o
r
ith
m
s
f
o
u
n
d
id
en
tic
al
s
o
lu
tio
n
s
.
Ho
wev
er
,
th
e
W
elsh
-
Po
well
alg
o
r
ith
m
f
o
u
n
d
b
e
tter
s
o
lu
tio
n
s
in
2
3
ca
s
es.
I
n
co
m
p
ar
is
o
n
,
t
h
e
DSatu
r
alg
o
r
ith
m
f
o
u
n
d
b
etter
s
o
lu
tio
n
s
in
o
n
l
y
n
in
e
ca
s
es,
b
u
t
f
o
r
g
r
ap
h
s
with
a
lar
g
er
n
u
m
b
er
o
f
v
er
tices a
n
d
ed
g
es (
in
th
e
s
ec
o
n
d
h
al
f
o
f
t
h
e
g
r
ap
h
s
et)
.
Fo
r
all
o
th
er
ca
s
es
(
f
o
r
t
h
is
s
et
o
f
g
r
ap
h
s
)
,
b
o
th
alg
o
r
ith
m
s
f
o
u
n
d
id
en
tical
s
o
lu
tio
n
s
.
I
n
th
e
s
ec
o
n
d
h
alf
o
f
th
e
s
et
o
f
g
r
ap
h
s
(
2
4
–
4
6
)
th
is
tr
e
n
d
ch
an
g
es.
Fo
r
t
h
ese
g
r
ap
h
s
,
in
o
n
ly
f
iv
e
o
f
th
e
ca
s
es
(
g
r
a
p
h
s
3
2
,
3
3
,
3
4
,
4
2
,
an
d
4
4
)
th
e
DSatu
r
alg
o
r
ith
m
f
o
u
n
d
b
etter
s
o
l
u
tio
n
s
th
an
th
e
W
elsh
-
Po
well
alg
o
r
ith
m
.
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
.
4
,
Au
g
u
s
t
20
25
:
3
8
6
7
-
3875
3872
Fig
u
r
e
4
.
B
est s
o
lu
tio
n
s
f
o
u
n
d
b
y
b
o
th
alg
o
r
ith
m
s
Fig
u
r
e
5
.
Dif
f
e
r
en
ce
in
co
lo
r
s
b
etwe
en
th
e
DSatu
r
alg
o
r
ith
m
an
d
th
e
W
elsh
-
Po
well
alg
o
r
ith
m
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
I
n
co
n
tr
ast,
th
e
W
elsh
-
Po
well
alg
o
r
ith
m
f
o
u
n
d
b
etter
s
o
lu
tio
n
s
in
n
in
e
o
th
e
r
ca
s
es (
f
o
r
g
r
a
p
h
s
2
5
,
2
6
,
3
5
,
3
6
,
3
7
,
3
8
,
3
9
,
4
1
,
a
n
d
4
6
)
.
I
n
th
e
r
em
ain
in
g
n
in
e
ca
s
es
f
r
o
m
th
e
s
ec
o
n
d
h
alf
o
f
th
e
s
et
o
f
g
r
ap
h
s
,
b
o
th
alg
o
r
ith
m
s
f
o
u
n
d
i
d
en
tical
s
o
lu
tio
n
s
(
f
o
r
g
r
ap
h
s
2
4
,
2
7
,
2
8
,
2
9
,
3
0
,
3
1
,
4
0
,
4
3
,
a
n
d
4
5
)
.
Fro
m
th
e
ch
ar
t
in
Fig
u
r
e
5
,
wh
e
n
th
e
n
u
m
b
e
r
o
f
v
er
tices
in
th
e
g
r
a
p
h
in
c
r
ea
s
es,
r
esp
ec
tiv
ely
,
wh
en
th
e
n
u
m
b
er
o
f
e
d
g
es
in
th
e
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
A
n
a
n
a
lysi
s
b
etw
ee
n
th
e
Wels
h
-
P
o
w
ell
a
n
d
D
S
a
tu
r
a
lg
o
r
ith
ms fo
r
…
(
R
a
d
o
s
la
va
K
r
a
leva
)
3873
g
r
ap
h
in
cr
ea
s
es,
s
o
d
o
th
e
tim
es
wh
en
th
e
W
el
s
h
-
Po
well
al
g
o
r
ith
m
f
in
d
s
b
etter
s
o
lu
ti
o
n
s
.
T
h
is
tr
en
d
ca
n
b
e
s
ee
n
f
r
o
m
th
e
c
h
ar
t
i
n
Fig
u
r
e
5
,
w
h
ich
s
h
o
ws
th
e
d
if
f
er
en
ce
in
s
o
lu
tio
n
s
f
o
u
n
d
(
as
th
e
n
ee
d
ed
n
u
m
b
er
o
f
co
lo
r
s
)
b
etwe
en
b
o
th
DSatu
r
an
d
W
elsh
-
Po
well
alg
o
r
ith
m
s
.
Po
s
itiv
e
v
alu
es
s
h
o
w
b
etter
s
o
lu
tio
n
s
f
o
u
n
d
b
y
th
e
W
elsh
-
Po
well
alg
o
r
ith
m
.
So
,
n
eg
ativ
e
v
alu
es sh
o
w
b
ett
er
s
o
lu
tio
n
s
f
o
u
n
d
b
y
th
e
DSat
u
r
alg
o
r
ith
m
.
T
h
e
ch
ar
t
in
Fig
u
r
e
6
s
h
o
ws
th
e
ef
f
ec
t
o
f
i
n
cr
ea
s
in
g
th
e
s
ize
o
f
th
e
g
r
ap
h
s
(
i.e
.
,
in
cr
e
asin
g
th
e
n
u
m
b
er
o
f
v
er
tices
an
d
t
h
e
n
u
m
b
er
o
f
ed
g
es)
o
n
t
h
e
ex
ec
u
ti
o
n
tim
e
o
f
b
o
t
h
alg
o
r
ith
m
s
.
T
h
e
ex
ec
u
tio
n
tim
e
o
f
th
e
DSatu
r
alg
o
r
ith
m
is
g
r
ea
t
er
th
an
th
e
ex
ec
u
tio
n
tim
e
o
f
th
e
W
elsh
-
Po
well
alg
o
r
ith
m
,
r
ea
ch
in
g
u
p
t
o
a
m
in
u
te
f
o
r
g
r
a
p
h
s
with
a
lar
g
er
n
u
m
b
er
o
f
v
e
r
tices.
Fo
r
e
x
am
p
le,
f
o
r
g
r
ap
h
4
6
,
th
e
ex
ec
u
tio
n
tim
e
o
f
th
e
DSatu
r
alg
o
r
ith
m
is
7
9
s
ec
o
n
d
s
,
an
d
th
e
e
x
ec
u
tio
n
tim
e
o
f
th
e
W
elsh
-
Po
well
alg
o
r
ith
m
is
o
n
ly
6
.
5
s
ec
o
n
d
s
.
T
h
e
d
if
f
er
e
n
ce
b
etwe
en
th
ese
v
alu
es
is
ap
p
r
o
x
im
ately
1
2
t
im
es.
Fo
r
g
r
ap
h
s
with
f
ewe
r
v
er
tices
an
d
ed
g
es
(
f
r
o
m
th
e
f
ir
s
t
h
alf
o
f
th
e
g
r
a
p
h
s
et)
,
th
e
ex
ec
u
tio
n
tim
es
o
f
b
o
th
alg
o
r
ith
m
s
ar
e
s
m
aller
,
b
u
t
th
e
d
if
f
er
e
n
ce
is
ag
ain
m
an
y
tim
es lar
g
er
f
o
r
th
e
DSatu
r
alg
o
r
ith
m
.
4.
CO
NCLU
SI
O
N
I
n
th
is
r
esear
ch
,
an
an
al
y
s
is
b
etwe
en
two
alg
o
r
ith
m
s
W
elsh
-
Po
well
an
d
DSatu
r
,
u
s
ed
f
o
r
co
lo
r
in
g
s
p
ar
s
e
g
r
ap
h
s
,
was
p
r
esen
ted
.
So
m
e
ea
r
lier
w
o
r
k
in
t
h
e
f
ield
o
f
g
r
a
p
h
t
h
eo
r
y
,
r
elate
d
to
s
o
l
v
in
g
t
h
is
p
r
o
b
lem
,
was
also
d
is
cu
s
s
ed
.
T
wo
ap
p
r
o
x
im
ate
alg
o
r
ith
m
s
-
th
e
W
elsh
-
Po
well
alg
o
r
ith
m
an
d
th
e
DSatu
r
alg
o
r
ith
m
,
wer
e
im
p
lem
en
ted
an
d
an
aly
ze
d
.
T
h
e
g
lo
b
al
d
ec
lar
atio
n
s
o
f
d
if
f
er
en
t
p
ar
am
eter
s
u
s
ed
b
y
b
o
th
alg
o
r
ith
m
s
(
v
ar
iab
les
an
d
ar
r
ay
s
)
wer
e
s
h
o
wn
.
T
h
e
s
o
u
r
ce
co
d
e
o
f
th
e
ap
p
r
o
x
im
ate
alg
o
r
ith
m
s
was
p
r
esen
ted
an
d
an
aly
ze
d
.
Du
e
to
th
e
m
u
ltit
ask
in
g
wo
r
k
o
f
th
e
o
p
er
atin
g
s
y
s
tem
,
th
e
r
u
n
tim
e
o
f
b
o
th
alg
o
r
ith
m
s
was
r
e
-
ca
lcu
lated
as th
e
av
er
ag
e
o
f
f
iv
e
s
tar
ts
f
o
r
ea
ch
o
f
th
e
4
6
a
n
a
ly
ze
d
g
r
a
p
h
s
.
T
h
e
r
esu
lts
s
h
o
w
th
at
in
th
e
f
ir
s
t
h
alf
o
f
th
e
s
et
o
f
g
r
a
p
h
s
(
1
–
2
3
)
in
o
n
ly
f
iv
e
o
u
t
o
f
a
to
tal
o
f
2
3
ca
s
es,
th
e
W
elsh
-
Po
well
alg
o
r
ith
m
f
o
u
n
d
th
e
s
am
e
s
o
lu
tio
n
s
as
th
e
DSatu
r
alg
o
r
ith
m
.
Fo
r
all
o
t
h
er
ca
s
es,
b
o
th
alg
o
r
ith
m
s
f
o
u
n
d
id
en
tical
s
o
lu
tio
n
s
.
T
h
is
tr
en
d
ch
an
g
e
d
f
o
r
th
e
s
ec
o
n
d
h
al
f
o
f
th
e
g
r
ap
h
s
s
et
(
2
4
–
4
6
)
.
I
n
o
n
l
y
f
i
v
e
ca
s
es
o
f
th
e
g
r
a
p
h
s
s
tu
d
ied
,
th
e
DSatu
r
alg
o
r
ith
m
f
o
u
n
d
b
etter
s
o
lu
tio
n
s
th
an
t
h
e
W
elsh
-
Po
well
alg
o
r
ith
m
.
I
n
c
o
n
tr
as
t,
th
e
W
elsh
-
Po
well
alg
o
r
ith
m
f
o
u
n
d
b
etter
s
o
lu
tio
n
s
in
n
in
e
o
th
er
ca
s
es.
B
o
th
alg
o
r
ith
m
s
f
o
u
n
d
id
en
ti
ca
l
s
o
lu
tio
n
s
in
th
e
r
em
ain
in
g
n
in
e
ca
s
es
f
r
o
m
th
e
s
ec
o
n
d
h
alf
o
f
th
e
g
r
ap
h
s
s
et.
I
n
s
u
m
m
ar
y
,
b
o
t
h
alg
o
r
ith
m
s
f
o
u
n
d
i
d
e
n
tical
s
o
lu
tio
n
s
f
o
r
th
e
co
m
p
lete
s
et
o
f
g
r
a
p
h
s
in
o
n
ly
1
4
ca
s
es.
Ho
wev
er
,
th
e
W
elsh
-
Po
well
alg
o
r
ith
m
f
o
u
n
d
b
etter
s
o
lu
tio
n
s
in
2
3
o
th
e
r
ca
s
es.
I
n
co
m
p
ar
is
o
n
,
th
e
DSatu
r
alg
o
r
ith
m
f
o
u
n
d
b
ette
r
s
o
lu
tio
n
s
in
o
n
ly
n
in
e
ca
s
es,
b
u
t
f
o
r
g
r
ap
h
s
with
a
lar
g
er
n
u
m
b
er
o
f
v
e
r
tices
an
d
e
d
g
es.
W
h
en
th
e
n
u
m
b
e
r
o
f
v
er
tices
(
r
esp
ec
tiv
ely
,
t
h
e
n
u
m
b
er
o
f
e
d
g
es)
i
n
th
e
g
r
ap
h
i
n
cr
ea
s
es,
th
e
W
elsh
-
Po
well
a
lg
o
r
ith
m
s
tar
ts
to
f
in
d
b
etter
s
o
lu
tio
n
s
m
o
r
e
o
f
ten
th
an
th
e
DSatu
r
alg
o
r
ith
m
.
Ab
o
u
t
th
e
q
u
ality
o
f
th
e
s
o
lu
tio
n
s
g
en
er
a
t
ed
(
b
y
b
o
th
alg
o
r
ith
m
s
)
,
th
e
W
elsh
-
Po
well
alg
o
r
ith
m
f
o
u
n
d
b
etter
s
o
lu
tio
n
s
in
5
0
%
o
f
th
e
ca
s
es
(
2
3
in
to
tal)
.
So
,
th
e
DSatu
r
alg
o
r
ith
m
f
o
u
n
d
b
etter
s
o
lu
tio
n
s
in
o
n
ly
1
9
.
6
%
o
f
ca
s
es
(
9
in
to
tal)
.
I
n
th
e
r
em
ai
n
in
g
3
0
.
4
%
o
f
ca
s
es,
b
o
th
alg
o
r
ith
m
s
f
o
u
n
d
id
en
ti
ca
l
s
o
lu
tio
n
s
.
T
h
e
r
es
u
lts
s
h
o
w
th
e
ef
f
ec
t
o
f
in
c
r
ea
s
in
g
t
h
e
s
ize
o
f
th
e
g
r
ap
h
s
o
n
th
e
ex
ec
u
tio
n
tim
e
o
f
b
o
th
al
g
o
r
ith
m
s
.
T
h
e
ex
e
cu
tio
n
tim
e
o
f
th
e
DSatu
r
alg
o
r
ith
m
is
g
r
ea
ter
th
an
th
e
ex
ec
u
ti
o
n
tim
e
o
f
th
e
W
elsh
-
Po
well
alg
o
r
ith
m
,
r
e
ac
h
in
g
u
p
to
a
m
i
n
u
te
f
o
r
g
r
ap
h
s
with
a
la
r
g
er
n
u
m
b
er
o
f
v
er
tices.
Fo
r
g
r
ap
h
s
wit
h
f
ewe
r
v
e
r
tices
an
d
e
d
g
es
(
f
r
o
m
th
e
f
i
r
s
t
h
alf
o
f
th
e
g
r
ap
h
s
et)
,
th
e
ex
ec
u
tio
n
tim
es
o
f
b
o
th
alg
o
r
ith
m
s
ar
e
s
h
o
r
ter
,
b
u
t
th
e
d
if
f
er
en
ce
is
ag
ain
m
an
y
tim
es
lar
g
er
f
o
r
th
e
DSatu
r
alg
o
r
ith
m
.
T
h
e
s
tu
d
y
ca
n
b
e
c
o
n
tin
u
ed
b
y
p
er
f
o
r
m
in
g
a
c
o
m
p
ar
at
iv
e
an
aly
s
is
o
f
th
e
p
er
f
o
r
m
an
ce
o
f
b
o
th
al
g
o
r
ith
m
s
if
th
e
an
al
y
ze
d
g
r
ap
h
s
ar
e
r
ep
r
esen
ted
b
y
o
th
er
d
ata
s
tr
u
ctu
r
es,
s
u
ch
as
ad
jace
n
cy
lis
ts
.
F
UNDING
I
NF
O
R
M
A
T
I
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N
T
h
is
r
esear
ch
was c
o
n
d
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cte
d
with
o
u
t f
in
an
cial
s
u
p
p
o
r
t
f
r
o
m
ex
ter
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s
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r
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s
.
AUTHO
R
CO
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B
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NS ST
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M
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h
is
jo
u
r
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th
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C
o
n
tr
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u
to
r
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o
les
T
ax
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y
(
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to
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al
au
th
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th
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r
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h
ip
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p
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an
d
f
ac
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co
llab
o
r
atio
n
.
Na
m
e
o
f
Aut
ho
r
C
M
So
Va
Fo
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Vi
Su
P
Fu
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s
lav
a
Kr
alev
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✓
✓
✓
✓
✓
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✓
✓
✓
✓
Velin
Kr
alev
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
T
o
m
a
Katsar
s
k
i
✓
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
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8
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I
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&
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,
Vo
l.
15
,
No
.
4
,
Au
g
u
s
t
20
25
:
3
8
6
7
-
3875
3874
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C
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:
M
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d
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g
y
So
:
So
f
t
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Va
:
Va
l
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d
a
t
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n
Fo
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Fo
r
mal
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n
v
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v
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P
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Fu
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Au
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s
s
tate
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est.
DATA AV
AI
L
AB
I
L
I
T
Y
T
h
e
d
ata
th
at
s
u
p
p
o
r
t
th
e
f
in
d
in
g
s
o
f
th
is
s
tu
d
y
ar
e
a
v
aila
b
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f
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m
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co
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esp
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eq
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est.
RE
F
E
R
E
NC
E
S
[
1
]
J.
H
u
a
n
g
,
X
.
G
e
n
g
,
S
.
L
i
,
a
n
d
Z.
Z
h
o
u
,
“
O
n
sp
e
c
t
r
a
l
e
x
t
r
e
m
a
o
f
g
r
a
p
h
s
w
i
t
h
g
i
v
e
n
o
r
d
e
r
a
n
d
d
i
s
so
c
i
a
t
i
o
n
n
u
mb
e
r
,
”
D
i
scr
e
t
e
Ap
p
l
i
e
d
M
a
t
h
e
m
a
t
i
c
s
,
v
o
l
.
3
4
2
,
p
p
.
3
6
8
–
3
8
0
,
Ja
n
.
2
0
2
4
,
d
o
i
:
1
0
.
1
0
1
6
/
j
.
d
a
m.
2
0
2
3
.
1
0
.
0
0
1
.
[
2
]
X
.
Z
h
a
n
g
,
S
.
Z
h
a
n
g
,
C
.
Y
e
,
a
n
d
B
.
Y
a
o
,
“
G
r
a
p
h
i
c
l
a
t
t
i
c
e
s
m
a
d
e
b
y
g
r
a
p
h
f
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l
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c
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s
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t
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p
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b
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d
c
o
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g
s
o
f
t
o
p
o
l
o
g
i
c
a
l
c
o
d
i
n
g
,
”
D
i
scr
e
t
e
A
p
p
l
i
e
d
Ma
t
h
e
m
a
t
i
c
s
,
v
o
l
.
3
3
6
,
p
p
.
3
7
–
4
6
,
S
e
p
.
2
0
2
3
,
d
o
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:
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0
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1
0
1
6
/
j
.
d
a
m.
2
0
2
3
.
0
3
.
0
2
3
.
[
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
,
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o
.
3
,
p
p
.
5
6
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–
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8
2
,
2
0
2
4
,
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o
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:
1
0
.
1
4
3
1
7
/
j
a
mi
.
2
0
2
4
.
5
6
7
.
[
4
]
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
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s
u
p
e
r
a
n
t
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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
,
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o
v
.
2
0
2
0
,
d
o
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:
1
0
.
3
3
9
0
/
s
y
m1
2
1
1
1
8
4
3
.
[
5
]
Z.
C
a
o
,
Q
.
Zh
a
n
g
,
J.
G
u
a
n
g
,
S
.
W
u
,
Z.
H
u
,
a
n
d
J.
Li
u
,
“
S
e
m
a
n
t
i
c
T
o
p
o
L
o
o
p
:
S
e
ma
n
t
i
c
l
o
o
p
c
l
o
su
r
e
w
i
t
h
3
D
t
o
p
o
l
o
g
i
c
a
l
g
r
a
p
h
b
a
s
e
d
o
n
q
u
a
d
r
i
c
-
l
e
v
e
l
o
b
j
e
c
t
m
a
p
,
”
I
EEE
R
o
b
o
t
i
c
s
a
n
d
A
u
t
o
m
a
t
i
o
n
L
e
t
t
e
rs
,
v
o
l
.
9
,
n
o
.
5
,
p
p
.
4
2
5
7
–
4
2
6
4
,
M
a
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