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r
o
b
lem
o
f
co
n
s
tr
u
ct
in
g
R
SMT
is
s
h
o
w
n
to
b
e
N
P
co
m
p
lete
b
y
Gar
e
y
an
d
J
o
h
n
s
o
n
[
2
]
.
T
h
e
r
atio
o
f
t
h
e
le
n
g
t
h
o
f
r
ec
til
i
n
ea
r
v
er
s
io
n
o
f
Min
i
m
u
m
Sp
an
n
in
g
T
r
ee
(
R
MST
)
to
th
at
o
f
R
SMT
is
p
r
o
v
ed
to
b
e
≤
3
/2
b
y
H
w
a
n
g
[
3
]
.
Kh
a
n
g
a
n
d
R
o
b
in
s
p
r
o
p
o
s
ed
I
ter
ated
1
-
Stein
er
(
I
1
S)
alg
o
r
ith
m
th
a
t
iter
ativ
e
l
y
ad
d
s
a
Ste
in
er
p
o
in
t
th
at
r
es
u
lts
i
n
co
s
t
r
ed
u
ctio
n
an
d
a
B
atch
ed
v
er
s
io
n
(
B
atc
h
e
d
I
ter
ated
1
-
Stein
er
al
g
o
r
ith
m
)
w
h
er
e
a
g
r
o
u
p
o
f
Stein
er
p
o
in
ts
ar
e
ad
d
ed
d
u
r
in
g
ea
ch
iter
atio
n
[
4
]
.
B
o
r
ah
,
Ow
e
n
s
a
n
d
I
r
w
in
p
r
esen
ted
an
ed
g
e
-
b
ased
h
e
u
r
is
tic
alg
o
r
ith
m
w
h
ic
h
i
n
itiall
y
co
n
s
tr
u
cts
a
M
in
i
m
u
m
Sp
an
n
i
n
g
T
r
ee
an
d
tr
an
s
f
o
r
m
s
it
i
n
to
a
R
SMT
b
y
it
er
ati
v
el
y
co
n
n
ec
ti
n
g
a
p
o
i
n
t
to
th
e
en
c
l
o
s
in
g
r
ec
ta
n
g
u
lar
la
y
o
u
t
o
f
t
h
e
v
i
s
ib
le
ed
g
e
i
n
t
h
e
M
ST
[
5
]
.
Z
h
o
u
p
r
o
p
o
s
ed
a
R
ec
tili
n
ea
r
Sp
an
n
i
n
g
g
r
ap
h
(
R
SG)
alg
o
r
ith
m
[
6
]
w
h
ich
ap
p
li
es
B
o
r
ah
et
al
ed
g
e
b
ased
h
eu
r
is
tic
al
g
o
r
ith
m
5
on
Sp
an
n
in
g
T
r
ee
co
n
s
tr
u
cted
f
r
o
m
th
e
Z
h
o
u
at
al
s
p
a
n
n
in
g
g
r
ap
h
al
g
o
r
ith
m
w
h
ich
was
co
n
s
tr
u
cted
b
y
co
n
n
ec
ti
n
g
ea
ch
p
o
in
t to
t
h
e
n
ea
r
est p
o
in
t in
ei
g
h
t o
ctal
r
eg
i
o
n
s
[
7
]
.
Gr
if
f
it
h
,
J
ef
f
,
et
al
p
r
o
p
o
s
ed
a
v
ar
ian
t
o
f
B
I
1
S
u
s
i
n
g
d
y
n
a
m
ic
MST
u
p
d
ate
s
ch
e
m
e
w
h
e
r
e
a
p
o
in
t
is
co
n
n
ec
ted
to
th
e
n
ea
r
est
p
o
in
ts
i
n
ei
g
h
t
o
cta
n
ts
a
n
d
th
e
lo
n
g
e
s
t
ed
g
e
is
r
e
m
o
v
ed
in
t
h
e
f
o
r
m
ed
lo
o
p
[
8
]
.
K
h
a
ng
,
M
a
n
do
iu
a
n
d
Z
el
i
k
ov
s
k
y
pr
o
po
s
ed
a
b
atc
h
ed
v
e
r
s
i
o
n
o
f
g
r
ee
d
y
t
r
i
p
le
c
o
n
t
r
ac
ti
o
n
al
g
or
i
t
h
m
[
9
]
ca
lled
B
atch
ed
G
r
ee
d
y
Al
g
or
i
t
h
m
w
h
e
r
e
R
SMT
is
co
n
s
tr
u
cted
b
y
iter
ati
v
el
y
ad
d
in
g
a
b
at
c
h
o
f
t
r
i
p
les
(
o
p
t
i
m
al
f
u
ll
Ste
i
n
er
t
r
ee
f
o
r
a
s
et
o
f
3
po
i
n
ts
wi
t
h
all
t
h
e
po
i
n
ts
i
n
th
e
lea
v
e
s
p
o
s
iti
o
n
)
[
1
0
]
.
W
o
ng
,
Yiu
-
C
hu
n
g
a
n
d
C
h
u
p
r
o
p
o
s
ed
a
F
ast
L
o
o
k
-
Up
T
ab
le
b
ased
alg
o
r
ith
m
w
i
th
a
p
r
e
-
co
m
p
u
ted
ta
b
le
f
o
r
co
n
s
tr
u
cti
n
g
R
SMT
f
o
r
N≤
9
.
Fo
r
N>
9
,
a
n
et
b
r
ea
k
in
g
al
g
o
r
ith
m
is
iter
at
iv
el
y
u
s
ed
u
n
til
N<
9
an
d
th
e
p
r
e
-
co
m
p
u
ted
tab
le
ca
n
b
e
u
s
ed
[
1
1
]
.
R
SMT
w
a
s
also
co
n
s
tr
u
cted
b
y
co
n
n
ec
t
in
g
t
h
e
tr
ee
s
th
a
t
h
av
e
b
ee
n
co
n
s
tr
u
cted
f
o
r
t
h
e
co
m
p
u
ted
clu
s
ter
s
o
f
g
i
v
e
n
p
o
in
ts
[
1
2
]
.
T
h
e
ex
is
tin
g
alg
o
r
ith
m
s
f
o
r
th
e
co
n
s
tr
u
ct
io
n
o
f
R
SMT
h
av
e
b
ee
n
ex
ten
s
i
v
el
y
s
u
r
v
e
y
ed
[
1
3
]
.
2.
RE
S
E
ARCH
M
E
T
H
O
D
T
h
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
w
o
r
k
s
b
y
d
r
a
w
in
g
in
cr
e
m
e
n
tal
f
o
u
r
lin
e
s
eg
m
en
ts
th
r
o
u
g
h
ea
ch
an
d
ev
er
y
p
o
in
t.
T
h
e
ed
g
es
ar
e
iter
ati
v
el
y
ad
d
ed
w
h
e
n
t
w
o
li
n
e
s
eg
m
e
n
ts
i
n
ter
s
ec
t.
Fo
r
ea
ch
o
f
th
e
ed
g
e,
t
w
o
L
-
s
h
ap
ed
la
y
o
u
t
s
ca
n
b
e
id
en
ti
f
ied
.
T
h
e
L
-
s
h
ap
ed
la
y
o
u
t
w
h
ic
h
h
as
an
o
v
er
lap
w
it
h
o
th
er
ed
g
es
s
h
o
u
ld
b
e
cle
v
er
l
y
s
elec
ted
to
r
ed
u
ce
th
e
o
v
er
all
co
s
t
o
f
th
e
R
SMT
.
I
f
a
d
ec
is
io
n
i
n
s
elec
ti
n
g
a
la
y
o
u
t
ca
n
n
o
t
m
e
m
ad
e
o
r
i
f
b
o
th
ar
e
o
f
s
a
m
e
le
n
g
t
h
,
t
h
e
p
r
o
ce
s
s
o
f
ad
d
in
g
t
h
e
ed
g
e
to
th
e
R
S
MT
w
ill
b
e
d
ela
y
ed
u
n
til
p
r
o
p
er
d
ec
is
io
n
ca
n
n
o
t
b
e
m
ad
e.
T
h
e
en
h
a
n
ce
m
en
t
to
Au
g
m
e
n
t
L
in
e
Se
g
m
e
n
t
B
ased
(
A
L
SB
)
A
l
g
o
r
ith
m
is
ca
r
r
ie
d
o
u
t
b
y
d
o
u
b
lin
g
th
e
s
ize
o
f
li
n
e
s
eg
m
e
n
ts
in
ea
ch
iter
atio
n
u
n
le
s
s
t
h
e
y
cr
o
s
s
th
e
i
n
ter
s
ec
tio
n
p
o
in
t
[
1
4
]
.
I
f
th
e
y
cr
o
s
s
in
ter
s
ec
tio
n
p
o
in
t,
th
e
le
n
g
th
o
f
th
e
l
i
n
e
s
e
g
m
e
n
t
s
w
ill
b
e
r
ed
u
ce
d
b
ac
k
to
th
e
p
r
ev
io
u
s
v
a
lu
e
an
d
au
g
m
e
n
tin
g
s
tar
ts
w
ill
v
al
u
e
o
f
1
.
T
h
e
p
r
o
c
ed
u
r
e
ca
r
r
ied
o
u
t is as f
o
llo
w
s
:
-
1.
I
d
en
tify
th
e
b
o
u
n
d
ar
y
-
B
o
u
n
d
ar
y
i
s
co
m
p
u
ted
b
y
id
en
ti
f
y
in
g
th
e
m
i
n
i
m
u
m
a
n
d
m
a
x
i
m
u
m
x
a
n
d
y
v
alu
e
s
.
T
h
e
len
g
t
h
o
f
t
h
e
li
n
e
s
eg
m
e
n
ts
ar
e
i
n
cr
e
m
e
n
ted
u
n
til
th
e
y
to
u
c
h
t
h
e
b
o
u
n
d
ar
y
o
r
u
n
ti
l
th
e
R
SMT
is
co
n
s
tr
u
cted
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
J
E
C
E
Vo
l.
7
,
No
.
3
,
J
u
n
e
2
0
1
7
:
1
2
6
2
–
1
2
6
7
1264
2.
Au
g
m
e
n
t
t
h
e
li
n
e
s
e
g
m
e
n
ts
-
T
h
e
len
g
t
h
o
f
t
h
e
li
n
e
s
eg
m
e
n
ts
ar
e
d
o
u
b
led
ea
ch
ti
m
e
u
n
t
il
it
cr
o
s
s
es
th
e
i
n
ter
s
ec
tio
n
p
o
in
t
o
f
a
n
y
t
w
o
li
n
e
s
e
g
m
en
t
s
else
th
e
le
n
g
th
o
f
all
t
h
e
li
n
e
s
e
g
m
en
t
s
w
i
ll
b
e
s
et
to
th
e
p
r
ev
io
u
s
v
al
u
es a
n
d
ag
ai
n
s
tar
ts
au
g
m
e
n
ti
n
g
w
i
th
t
h
e
s
te
p
_
s
ize
o
f
o
n
e.
3.
C
o
n
s
tr
u
ct
R
SMT
-
R
SMT
w
ill
b
e
co
n
s
tr
u
cted
b
y
i
n
cr
e
m
en
tall
y
ad
d
in
g
ed
g
es
w
h
e
n
t
w
o
li
n
e
s
eg
m
e
n
ts
i
n
ter
s
ec
t
a
n
d
i
f
th
at
ed
g
e
d
o
es
n
o
t
f
o
r
m
a
lo
o
p
in
t
h
e
p
ar
tiall
y
co
n
s
tr
u
cted
R
SM
T
.
A
d
d
in
g
an
ed
g
e
r
eq
u
ir
es
s
elec
ti
n
g
o
n
e
o
f
t
h
e
t
w
o
L
-
S
h
ap
ed
la
y
o
u
t
w
h
ic
h
r
es
u
lt
s
i
n
co
s
t
r
ed
u
cti
o
n
.
I
f
b
o
t
h
th
e
ed
g
es
ar
e
o
f
s
a
m
e
le
n
g
th
,
th
en
b
o
th
ed
g
e
s
w
ill
b
e
m
ar
k
ed
as
te
m
p
o
r
ar
y
ed
g
es
u
n
til
a
d
ec
is
io
n
ca
n
b
e
m
ad
e.
Fin
al
l
y
w
h
e
n
n
o
m
o
r
e
ed
g
e
s
ca
n
b
e
ad
d
ed
an
d
if
te
m
p
o
r
ar
y
ed
g
e
s
ex
is
t
t
h
e
n
t
h
e
L
-
s
h
ap
ed
la
y
o
u
t
s
ar
e
ch
ec
k
ed
f
o
r
o
v
er
lap
w
i
th
t
h
e
co
n
s
tr
u
cted
R
S
MT
an
d
th
e
co
r
r
esp
o
n
d
in
g
L
a
y
o
u
t
w
ill b
e
ad
d
ed
.
2
.
1
.
Alg
o
rit
h
m
T
h
e
i
m
p
r
o
v
ed
AL
SB
al
g
o
r
ith
m
ta
k
es
as
an
i
n
p
u
t
a
s
et
o
f
p
o
in
t
s
an
d
co
m
p
u
tes
t
h
e
R
S
MT
alo
n
g
w
it
h
its
co
s
t o
r
to
tal
len
g
th
.
n
---
n
u
m
b
er
o
f
p
o
in
ts
C
u
r
r
en
t_
len
g
t
h
=0
s
tep
_
s
ize=
0
w
h
ile
(
n
u
m
_
ed
g
es<
n
-
1
)
b
eg
in
f
o
r
i=1
to
n
-
-
-
d
o
in
p
ar
allel
p
r
ev
io
u
s
_
len
g
t
h
=C
u
r
r
en
t_
le
n
g
th
o
f
li
n
e
s
eg
m
e
n
t
s
if
(
s
tep
_
s
ize=
=0
)
C
u
r
r
en
t_
len
g
t
h
=
p
r
ev
io
u
s
_
le
n
g
th
+1
s
tep
_
s
ize=
1
;
else
C
u
r
r
en
t_
len
g
t
h
=p
r
ev
io
u
s
_
len
g
th
+2
*
s
tep
_
s
ize
f
o
r
i=1
to
n
-
-
-
d
o
in
p
ar
allel
b
eg
in
if
(
t
w
o
lin
e
s
eg
m
e
n
ts
i
n
ter
s
ec
t
&
&
d
o
es
n
'
t
f
o
r
m
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I
J
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C
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I
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N:
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A
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ith
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1265
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h
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m
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ter
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t.
3.
RE
SU
L
T
S
A
ND
AN
AL
Y
SI
S
T
h
e
p
r
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s
ed
alg
o
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ith
m
h
as
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n
i
m
p
le
m
e
n
ted
in
C
.
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g
u
r
e
3
s
h
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w
s
t
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at
t
h
e
i
m
p
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v
ed
AL
SB
alg
o
r
ith
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w
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g
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o
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im
p
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m
e
n
t
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t
h
e
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o
r
ith
m
w
i
th
r
esp
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t
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m
b
e
r
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ti
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es
t
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li
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m
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e
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g
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ted
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s
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et
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o
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t
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th
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t
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ated
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t
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s
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ted
,
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o
o
d
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n
ca
n
b
e
id
en
ti
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ied
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ab
le
1
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em
o
n
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tr
at
e
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th
e
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s
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c
ti
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y
t
h
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i
m
p
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alg
o
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ith
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.
Fig
u
r
e
3
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C
o
m
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ar
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o
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w
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n
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an
d
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m
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v
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o
r
ith
m
T
ab
le
1
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o
m
p
ar
is
o
n
o
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th
e
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s
t o
f
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M
ST
,
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s
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g
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,
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ar
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e
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ab
le
2
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th
e
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etails
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m
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t c
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r
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<
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3
s
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lt
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o
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tain
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ap
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o
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a
lg
o
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it
h
m
o
n
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ar
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cted
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m
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m
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m
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ted
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r
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n
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ch
cir
cu
it is
a
s
s
h
o
w
n
i
n
T
ab
le
3
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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N
:
2
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8
8
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8708
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2
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ar
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8
8
41
T
ab
le
3
: A
p
p
licatio
n
o
f
i
m
p
r
o
v
ed
AL
SB
alg
o
r
ith
m
o
n
I
B
M
I
SP
D0
8
b
en
ch
m
ar
k
C
i
r
c
u
i
t
#
o
f
n
e
t
s
T
o
t
a
l
R
S
M
T
l
e
n
g
t
h
o
f
a
l
l
n
e
t
s
M
a
x
R
S
M
T
l
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n
g
t
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f
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t
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n
t
h
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c
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r
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u
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t
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b
m
0
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6
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3
3
3
3
4.
CO
NCLU
SI
O
N
T
h
e
p
r
o
p
o
s
ed
im
p
r
o
v
ed
AL
S
B
alg
o
r
ith
m
p
r
o
v
id
es
g
o
o
d
im
p
r
o
v
e
m
e
n
t
o
v
er
t
h
e
AL
SB
alg
o
r
ith
m
i
n
ter
m
s
o
f
n
u
m
b
er
o
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lin
e
s
e
g
m
en
t
i
n
cr
e
m
e
n
t
a
n
d
co
s
t
r
ed
u
ct
io
n
.
T
h
e
alg
o
r
it
h
m
w
as
a
ls
o
e
f
f
icien
tl
y
test
ed
o
n
I
B
M
I
SP
D
0
8
b
en
ch
m
ar
k
as
s
h
o
w
n
i
n
tab
le
3
.
Fu
tu
r
e
e
f
f
o
r
ts
w
o
u
ld
b
e
d
ir
ec
ted
to
w
ar
d
s
f
u
r
th
er
co
s
t
r
ed
u
ctio
n
o
f
th
e
tr
ee
an
d
to
i
m
p
le
m
e
n
t t
h
e
ab
o
v
e
alg
o
r
it
h
m
o
n
FP
GA
to
i
m
p
r
o
v
e
th
e
p
er
f
o
r
m
a
n
ce
.
RE
F
E
R
E
NC
E
S
[1
]
Ha
n
a
n
,
M
a
u
rice
,
“
On
S
tein
e
r'
s
P
r
o
b
lem
w
it
h
Re
c
ti
li
n
e
a
r
Dista
n
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e
”
,
S
IAM
J
o
u
rn
a
l
o
n
A
p
p
l
ied
M
a
t
h
e
ma
ti
c
s
,
1
4
.
2
(1
9
6
6
):
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5
5
-
2
6
5
[2
]
G
a
r
e
y
,
M
ich
a
e
l
R,
Da
v
id
S
.
Jo
h
n
so
n
,
“
T
h
e
Re
c
ti
li
n
e
a
r
S
tei
n
e
r
tr
e
e
P
r
o
b
lem
is
NP
-
C
o
m
p
lete
”,
S
I
AM
J
o
u
rn
a
l
o
n
Ap
p
li
e
d
M
a
t
h
e
ma
ti
c
s
,
3
2
.
4
(
1
9
7
7
):
8
2
6
-
8
3
4
.
[3
]
Hw
a
n
g
,
F
ra
n
k
K,
“
On
S
tein
e
r
M
in
im
a
l
tree
s
w
it
h
Re
c
ti
li
n
e
a
r
Dis
tan
c
e
”
,
S
IAM
J
o
u
rn
a
l
o
n
Ap
p
li
e
d
M
a
th
e
ma
ti
c
s
,
3
0
.
1
(
1
9
7
6
):
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0
4
-
1
1
4
.
[4
]
Ka
h
n
g
,
A
n
d
re
w
B.
,
Ga
b
riel
Ro
b
in
s,
“
A
Ne
w
Clas
s
o
f
Itera
ti
v
e
S
tein
e
r
tree
H
e
u
risti
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s
w
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h
G
o
o
d
P
e
rf
o
rm
a
n
c
e
”
,
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mp
u
ter
-
Ai
d
e
d
De
sig
n
o
f
In
teg
r
a
ted
Circ
u
i
ts
a
n
d
S
y
ste
ms
,
IEE
E
T
ra
n
sa
c
ti
o
n
s
on
1
1
.
7
(
1
9
9
2
):
8
9
3
-
9
0
2
.
[5
]
Bo
ra
h
,
M
a
n
ji
t,
R
o
b
e
rt
M
i
c
h
a
e
l
Ow
e
n
s,
M
a
ry
J
a
n
e
Ir
w
in
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An
Ed
g
e
-
b
a
se
d
He
u
risti
c
f
o
r
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te
in
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r
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ti
n
g
”,
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mp
u
ter
-
Ai
d
e
d
De
sig
n
o
f
In
teg
r
a
ted
Circ
u
i
ts
a
n
d
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y
ste
ms
,
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E
T
ra
n
sa
c
ti
o
n
s o
n
1
3
.
1
2
(1
9
9
4
):
1
5
6
3
-
1
5
6
8
.
[6
]
Zh
o
u
,
Ha
i,
“
Ef
fi
c
ien
t
S
tein
e
r
re
e
Co
n
st
ru
c
ti
o
n
b
a
se
d
o
n
S
p
a
n
n
i
n
g
Gr
a
p
h
s”,
P
r
o
c
e
e
d
in
g
s
o
f
th
e
2
0
0
3
i
n
tern
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ti
o
n
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l
s
y
m
p
o
siu
m
o
n
P
h
y
sic
a
l
d
e
sig
n
.
ACM,
2
0
0
3
.
[7
]
Zh
o
u
,
Ha
i,
Na
re
n
d
ra
S
h
e
n
o
y
,
W
il
li
a
m
Nic
h
o
ll
s,
“
Ef
f
icie
n
t
M
in
imu
m
S
p
a
n
n
in
g
tre
e
C
o
n
stru
c
ti
o
n
wit
h
o
u
t
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la
u
n
a
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ria
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g
u
l
a
ti
o
n
”
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P
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th
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1
A
sia
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n
d
S
o
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t
h
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if
ic De
sig
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u
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ti
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n
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n
f
e
re
n
c
e
.
A
C
M
,
2
0
0
1
.
[8
]
G
ri
ff
it
h
,
Je
ff
,
e
t
a
l,
“
Clo
sin
g
th
e
g
a
p
:
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a
r
-
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ti
m
a
l
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tein
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r
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i
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ly
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ial
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m
e
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mp
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ter
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d
e
d
De
sig
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o
f
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ra
te
d
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it
s
a
n
d
S
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ste
ms
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T
ra
n
sa
c
ti
o
n
s o
n
1
3
.
1
1
(1
9
9
4
):
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3
5
1
-
1
3
6
5
.
[9
]
Ka
h
n
g
,
A
n
d
re
w
B.
,
Io
n
I.
M
a
n
d
o
iu
,
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lex
a
n
d
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r
Z.
Zelik
o
v
sk
y
,
“
Hig
h
ly
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c
a
l
a
b
le
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g
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rit
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ms
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r
Rec
ti
li
n
e
a
r
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n
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ti
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e
a
r
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tein
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r
tre
e
s”
,
De
sig
n
A
u
to
m
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ti
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n
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o
n
f
e
re
n
c
e
,
2
0
0
3
.
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ro
c
e
e
d
i
n
g
s
o
f
th
e
A
S
P
-
DA
C
2
0
0
3
.
A
sia
a
n
d
S
o
u
t
h
P
a
c
if
ic
.
IEE
E,
2
0
0
3
.
[1
0
]
Zelik
o
v
sk
y
,
A
le
x
a
n
d
e
r
Z,
“
A
n
1
1
/6
-
A
p
p
r
o
x
im
a
ti
o
n
A
lg
o
rit
h
m
f
o
r
th
e
Ne
tw
o
rk
S
tein
e
r
P
ro
b
lem
”,
Al
g
o
rith
mic
a
,
9
.
5
(1
9
9
3
):
4
6
3
-
4
7
0
.
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I
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r
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ith
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1
]
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o
n
g
,
Yiu
-
Ch
u
n
g
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C
h
ris
Ch
u
,
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A
S
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la
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le
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n
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ra
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g
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L
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sig
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L
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-
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T
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ter
n
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ti
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l
S
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m
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m
o
n
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2
0
0
8
.
[1
2
]
V
a
n
i,
V
.
,
G
.
R.
P
ra
sa
d
,
“
Al
g
o
rit
h
m
fo
r
th
e
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o
n
stru
c
ti
o
n
o
f
Rec
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n
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r
S
tein
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r
M
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m
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b
y
id
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n
g
th
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Clu
ste
rs
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f
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in
ts
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f
o
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ti
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m
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n
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In
tern
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ti
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l
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n
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n
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2
0
1
4
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3
]
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a
n
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ra
sa
d
,
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e
rf
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n
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A
n
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m
s
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o
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c
ti
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o
f
Re
c
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li
n
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a
r
S
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r
M
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m
tree
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c
t
2
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1
3
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6
1
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8
.
[1
4
]
V
a
n
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ra
sa
d
,
“
A
u
g
me
n
te
d
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in
e
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me
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a
se
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g
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rit
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m
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r
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n
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c
ti
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g
Rec
ti
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n
e
a
r
S
tein
e
r
M
in
imu
m
t
re
e
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r
o
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d
in
g
s o
f
IEE
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n
tern
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ti
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n
d
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e
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tro
n
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ste
m
s
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0
1
6
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in
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re
ss
.
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5
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T
h
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Evaluation Warning : The document was created with Spire.PDF for Python.