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Th
e P
hysic
al P
lacem
en
t P
rob
lem
in
In
teg
rate
d C
ircu
its
Pro
f. K
urt
Ke
utz
er
EE
CS
Un
ive
rsit
y o
f C
ali
forn
ia
Be
rke
ley,
CA
Th
an
ks
to
Pro
f. A
. K
ah
ng
2K
urt
Keu
tzer
Sch
em
ati
c E
ntr
y D
esig
n F
low
sch
em
ati
ced
ito
r
netl
ist
Lib
rary
ph
ysic
al
desig
n
layo
ut
a b
s
q0 1
d
clk
a b
s
q0 1
d
clk lo
gic
sim
ula
tor
Des
ign
er d
esig
ns
the
circ
uit
on
nap
kin
s an
d b
lack
bo
ard
Gat
e-le
vel d
etai
ls o
f th
e
circ
uit
are
en
tere
d in
a
sch
emat
ic e
ntr
y to
ol
Vec
tors
are
gen
erat
ed t
o
veri
fy t
he
circ
uit
Wh
en lo
gic
is c
orr
ect
the
net
list
is p
asse
d o
ff t
o
ano
ther
gro
up
to
lay
ou
t
Au
tom
ated
pla
ce a
nd
ro
ute
too
ls c
reat
e la
you
t
RT
L D
esig
n F
low
RT
LS
yn
thesis
HD
L
netl
ist
log
ico
pti
miz
ati
on
netl
ist
Lib
rary
/m
od
ule
gen
era
tors
ph
ysic
al
desig
n
layo
ut
man
ual
desig
n
a b
s
q0 1
d
clk
a b
s
q0 1
d
clk
4K
urt
Keu
tzer
Th
e N
etl
ist
Wh
en
we f
inis
hed
syn
thesis
or
sch
em
ati
c e
ntr
y w
e
pro
du
ced
a n
etl
ist:
�F
un
cti
on
ally c
orr
ect
�C
orr
ect
tim
ing
rela
tive t
o t
he m
od
els
we u
sed
No
w w
e w
ish
to
tu
rn t
hat
into
a c
orr
ectl
y f
un
cti
on
ing
pla
cem
en
t
55
5
44
4
2
L A T C H
L A T C H
32
11
21
32
1
22
22
19
19
Ph
ys
ica
l D
es
ign
: O
ve
rall
Co
nc
ep
tua
l F
low
Rea
d N
etli
st
Init
ial
Pla
cem
ent
Pla
cem
ent
Imp
rovem
ent
Cost
Est
imati
on
Rou
tin
g R
egio
n
Def
init
ion
Glo
bal
Rou
tin
g
Inp
ut
Pla
cem
ent
Rou
tin
g
Ou
tpu
tC
om
pact
ion
/cle
an
-up
Rou
tin
g R
egio
n
Ord
erin
g
Det
ail
ed R
ou
tin
g
Cost
Est
imati
on
Rou
tin
g
Imp
rovem
ent
Wri
te L
ay
ou
t D
ata
base
Flo
orp
lan
nin
gF
loorp
lan
nin
g
Fo
rmu
lati
on
of
the P
lacem
en
t P
rob
lem
Giv
en
:
�A
ne
tlis
t o
f cell
s f
rom
a p
re-d
efi
ned
se
mic
on
du
cto
r li
bra
ry
�A
ma
the
mati
cal
exp
ressio
n o
f th
at
netl
ist
as a
vert
ex-,
ed
ge
-w
eig
hte
d g
rap
h
�C
on
str
ain
ts o
n p
in-l
ocati
on
s e
xp
ressed
as c
on
str
ain
ts o
n v
ert
ex
locati
on
s /
asp
ect
rati
o t
ha
t th
e p
lacem
en
t n
eed
s t
o f
it in
to
�O
ne
or
mo
re o
f th
e f
oll
ow
ing
: c
hip
-le
ve
l ti
min
g c
on
str
ain
ts,
a l
ist
of
cri
tical
ne
ts,
ch
ip-l
eve
l p
ow
er
co
nstr
ain
ts
Fin
d: �
Cell/v
ert
ex l
oca
tio
ns t
o m
inim
ize p
lace
men
t o
bje
cti
ve
su
bje
ct
toco
nstr
ain
ts
Ob
jec
tives
:
�m
inim
al
dela
y (
faste
st
cycle
tim
e)
�m
inim
al
are
a (
least
die
are
a/c
os
t
�o
ther
nic
eti
es
: e.g
. p
ow
er
7K
urt
Keu
tzer
Co
ns
tra
int-
dri
ve
n d
es
ign
Wh
en
we f
inis
hed
syn
thesis
or
sch
em
ati
c e
ntr
y w
e b
elieved
that
ou
r cir
cu
it m
et
its t
imin
g c
on
str
ain
ts b
ased
on
exp
ecte
d v
alu
es o
f in
terc
on
nect
dela
ys
We w
an
t d
ow
nstr
eam
ph
ysic
al d
esig
n t
oo
ls t
o h
on
or
tho
se
co
nstr
ain
ts!!
!
55
5
44
4
2
L A T C H
L A T C H
32
11
21
32
1
22
22
19
19
8K
urt
Keu
tzer
Resu
lts o
f P
lacem
en
t
A b
ad
pla
cem
ent
A g
oo
d p
lace
men
t
A. K
ah
ng
Wh
at’
s g
oo
d a
bo
ut
a g
oo
d p
lacem
en
t?W
hat’
s b
ad
ab
ou
t a b
ad
pla
cem
en
t?
9K
urt
Keu
tzer
Resu
lts o
f P
lacem
en
t
Bad
pla
ce
men
t c
au
ses r
ou
tin
g
co
ng
esti
on
res
ult
ing
in
:
•In
cre
ases
in
cir
cu
it a
rea (
co
st)
an
d w
irin
g
•L
on
ge
r w
ires � ���
mo
re c
ap
ac
itan
ce
�L
on
ger
dela
y
�H
igh
er
dyn
am
ic p
ow
er
dis
sip
ati
on
Go
od
pla
cm
en
t
•Cir
cu
it a
rea
(co
st)
an
d w
irin
g
decre
ases
•S
ho
rter
wir
es � ���
less c
ap
acit
an
ce
�S
ho
rter
dela
y
�L
ess d
yn
am
ic p
ow
er
dis
sip
ati
on
10
Ku
rt K
eu
tzer
Wh
y c
an
’t P
D o
be
y c
on
str
ain
ts?
Hig
h level (s
yn
thesis
, sch
em
ati
c e
ntr
y)
esti
mate
s o
f w
ire
dela
y m
ay b
e v
ery
naïv
e
As in
terc
on
nect
dela
y in
cre
ases r
ela
tive t
o g
ate
dela
y a
nd
beco
mes less p
red
icta
ble
it
may b
e im
po
ssib
le t
o p
lace
an
d r
ou
te a
cir
cu
it in
a w
ay t
hat
ho
no
rs in
itia
l co
nstr
aIn
ts
55
5
44
4
2
L A T C H
L A T C H
32
11
21
32
1
22
22
19
19
11
Ku
rt K
eu
tzer
Tim
ing
an
d P
D
How
do w
e get
[re
flec
t] t
he
del
ay n
um
ber
s on
th
e H
ow
do w
e get
[re
flec
t] t
he
del
ay n
um
ber
s on
th
e
gate
/in
terc
on
nec
t?gate
/in
terc
on
nec
t?
Sta
n C
how
Am
mocore
Andre
w B
. K
ahng U
CS
DM
ajid
Sarr
afz
adeh U
CLA
KK
: W
e h
ave
a f
ew
ch
oic
es a
bo
ut
wh
at
to d
o w
ith
th
e n
etl
ist
du
rin
g p
lacem
en
t:
•L
eave it
exactl
y t
he s
am
e
•Resiz
e g
ate
s a
nd
ad
d b
uff
ers
as w
e g
o
•R
e-f
acto
r/d
up
licate
lo
gic
55
5
44
4
2
L A T C H
L A T C H
32
11
21
32
1
22
22
19
19
12
Ku
rt K
eu
tzer
Tim
ing
an
d P
D -
2
How
do w
e get
[re
flec
t] t
he
del
ay n
um
ber
s on
th
e H
ow
do w
e get
[re
flec
t] t
he
del
ay n
um
ber
s on
th
e
gate
/in
terc
on
nec
t?gate
/in
terc
on
nec
t?
Sta
n C
how
Am
mocore
Andre
w B
. K
ahng U
CS
DM
ajid
Sarr
afz
adeh U
CLA
KK
: W
e h
ave
a f
ew
ch
oic
es a
bo
ut
wh
at
to d
o w
ith
th
e n
etl
ist
du
rin
g p
lacem
en
t:
•L
eave it
exactl
y t
he s
am
e –
sim
plify
ing
assu
mp
tio
n
•Resiz
e g
ate
s a
nd
ad
d b
uff
ers
as w
e g
o
•R
e-f
acto
r/d
up
licate
lo
gic
55
5
44
4
2
L A T C H
L A T C H
32
11
21
32
1
22
22
19
19
Go
rdia
n P
lacem
en
t F
low
Co
mp
lexit
ysp
ace:
O
(m)
ti
me:
Q
( m
1.5
log
2m
)F
inal p
lacem
en
t•s
tan
dard
cell •m
acro
-cell &
SO
G
Glo
ba
l O
pti
miz
ati
on
m
inim
iza
tio
n
o
f
wir
e len
gth
Part
itio
nin
g
of
the m
od
ule
set
an
d d
issecti
on
of
the p
lace
men
t re
gio
n
Fin
al
P
lacem
en
t
ad
ap
tio
no
f s
tyle
d
ep
en
den
t
co
nstr
ain
ts
mo
du
le c
oo
rdin
ate
s
po
sit
ion
co
nstr
ain
ts
mo
du
le
co
ord
inate
s
Reg
ion
s
w
ith
≤ ≤≤≤k
mo
du
les
Data
flo
w in
th
e p
lacem
en
t p
roced
ure
GO
RD
IAN
Gord
ian: A
Quadra
tic P
lacem
ent A
ppro
ach
•G
lob
al o
pti
miz
ati
on
:
so
lves a
seq
uen
ce o
f q
uad
rati
c
pro
gra
mm
ing
pro
ble
ms
•P
art
itio
nin
g:
en
forc
es t
he n
on
-overl
ap
co
nstr
ain
ts
15
Ku
rt K
eu
tzer
GO
RD
IAN
(q
uad
rati
c +
part
itio
nin
g)
Par
titi
on
and R
epla
ce
Init
ial
Pla
cem
ent
A. K
ah
ng
Go
rdia
n P
lacem
en
t F
low
J. K
lein
hau
s, G
. S
igl, F
. Jo
han
nes, K
. A
ntr
eic
h,
GO
RD
IAN
: V
LS
I P
lacem
en
t b
y Q
uad
rati
c
Pro
gra
mm
ing
an
d S
licin
g O
pti
miz
ati
on
, IE
EE
Tra
ns. o
n C
AD
, M
arc
h, 1991, p
p. 356 -
365
Lib
rary
Co
nta
ins f
or
each
cell:
�F
un
cti
on
al in
form
ati
on
: c
ell =
a *
b *
c
�T
imin
g in
form
ati
on
: fu
ncti
on
of
�in
pu
t sle
w
�in
trin
sic
dela
y
�o
utp
ut
cap
acit
an
ce
no
n-l
inear
mo
dels
used
in
tab
ula
r ap
pro
ach
�P
hysic
al fo
otp
rin
t (a
rea)
�P
ow
er
ch
ara
cte
risti
cs
Wir
e-l
oad
mo
dels
-fu
ncti
on
of
�B
lock s
ize
�W
irin
g
Lib
rary
Lib
rary
/m
od
ule
gen
era
tors
a b
s
q0 1
d
clk
Siz
e a
nd
asp
ect
rati
o o
f co
re d
ie
Netl
ist
->
100K
->
10M
cells f
rom
lib
rary
Lib
rary
, N
etl
ist,
an
d A
sp
ect
Rati
o
Sett
ing
up
Glo
bal O
pti
miz
ati
on
GO
RD
IAN
: G
lob
al P
lacem
en
t
We w
an
t to
op
tim
ize t
he d
ela
y o
f cri
tical
path
s
Inste
ad
we:
�o
pti
miz
e t
he s
um
of
sq
uare
s o
f n
et-
len
gth
s t
imes a
sta
tic w
eig
ht
Glo
bal p
lacem
en
t b
y q
uad
rati
c w
ire-l
en
gth
op
tim
izati
on
�P
rob
lem
is c
om
pu
tati
on
all
y t
racta
ble
an
d w
ell b
eh
aved
�G
lob
al co
nn
ecti
vit
y i
s c
on
sid
ere
d a
t all
sta
ges
�A
n i
ncre
asin
g n
um
ber
of
co
nstr
ain
ts i
s i
mp
osed
�G
lob
al p
lacem
en
t o
f m
od
ule
s i
s o
bta
ined
sim
ult
an
eo
usly
fo
r all
su
b-p
rob
lem
s
�N
o d
ep
en
den
ce o
n p
rocessin
g s
eq
uen
ce
Qu
ad
rati
c p
lacem
en
t clu
mp
s c
ell
s i
n c
en
ter
Part
itio
nin
g s
pre
ad
s c
ell
s a
nd
im
po
ses n
ew
co
nstr
ain
ts o
n f
urt
her
op
tim
izati
on
Intu
itiv
e f
orm
ula
tio
n
Giv
en
a s
eri
es o
f p
oin
ts x
1, x2, x3, …
xn
an
d a
co
nn
ecti
vit
y m
atr
ix C
descri
bin
g t
he c
on
necti
on
s
betw
een
th
em
(If
cij
= 1
th
ere
is a
co
nn
ecti
on
betw
een
xi an
d x
j)
Fin
d a
lo
cati
on
fo
r each
xjth
at
min
imiz
es t
he t
ota
l su
m o
f
all s
pri
ng
ten
sio
ns b
etw
een
each
pair
<xi, x
j>
xj
xi
Pro
ble
m h
as a
n o
bvio
us (
triv
ial)
so
luti
on
–w
hat
is it?
Imp
rovin
g t
he in
tuit
ive f
orm
ula
tio
n
To
avo
id t
he t
rivia
l so
luti
on
ad
d c
on
str
ain
ts:
Hx=
b
�T
hese m
ay b
e v
ery
natu
ral -
e.g
. en
dp
oin
ts (
pad
s)
To
in
teg
rate
th
e n
oti
on
of
``cri
tical n
ets
’’
�A
dd
weig
hts
w
ijto
nets
xj
xi
wij
-so
me
sp
rin
gs h
ave
mo
re t
en
sio
nsh
ou
ld p
ull
asso
cia
ted
vert
ices c
loser
x1
xn
wij
Mo
delin
g t
he N
et’
s W
ire L
en
gth
∑ ∑∑∑( (((
) )))( (((
) )))[ [[[
] ]]]y
yx
xL
Mu
vu
vv
uv
vv
− −−−+ +++
− −−−= === ← ←←←
22
mo
du
leu
(xv,y
v) )))
(xu
,yu
) ))))
,(
vuvu
η ηηηξ ξξξ
vup
in
vu lv
net
nod
e
x
yco
nn
ecti
on
to
o
ther
mo
du
les
( x
uv=
=
=
=
xu+ +++
uv
;ξ ξξξ
y uv
=
=
=
= y
u+ +++
y
)vu
Th
e l
en
gth
Lv
of
a n
et
v i
s m
easu
red
by t
he s
qu
are
d d
ista
nces f
rom
its
p
oin
ts t
o t
he n
et’
s c
en
ter
Wh
at
ab
ou
tm
ult
iterm
inal
nets
?
Wh
at
do
we r
eally w
an
t to
op
tim
ize?
Fo
r h
igh
-perf
orm
an
ce c
ircu
its, w
e w
an
t to
min
imiz
e lo
ng
est
path
th
rou
gh
netw
ork
•In
ad
dit
ion
to
to
tal w
ire len
gth
Ap
pro
xim
ate
dela
y m
inim
izati
on
by m
inim
izati
on
of
sq
uare
d w
ire len
gth
•P
en
alizes lo
ng
wir
es
Qu
ad
rati
c O
bje
cti
ve F
un
cti
on
Ob
jecti
ve f
un
cti
on
: w
eig
hte
d s
um
of
the s
qu
are
d n
et
len
gth
s
Wh
ere
N i
s n
ets
set,
wv
is n
et
weig
ht,
an
d L
vis
th
e m
ea
su
re o
f
net
len
gth
Can
re-f
orm
ula
te i
t in
term
s o
f b
lock c
oo
rdin
ate
s (
in o
ne 1
-D)
�xi
-lo
cati
on
s o
f vert
ices (
mo
du
les)
�w
ij -
ed
ge w
eig
hts
(n
et
weig
hts
)
�C
-th
e s
yste
m m
atr
ix
Gen
era
lize t
o 2
-D
νν
ν∈Ν
Φ=
∑L
w
Φ=
−=
∑2
Tij
ij
(x)
w(x
x)
0.5
xC
x
Φ=
−+
−=
+∑
22
TT
iji
ji
j(x
,y)
w[(
xx
)(y
y)
]0
.5x
Cx
0.5
yC
y
Fo
rmu
lati
ng
Op
tim
izati
on
Pro
ble
m
Fo
r sim
plicit
y, w
e’ll co
nsid
er
a 1
-D f
orm
ula
tio
n
Need
to
ad
d a
term
to
acco
un
t fo
r fi
xed
mo
du
les
Th
e v
ecto
r d
is c
on
str
ucte
d b
ased
co
ord
inate
s o
f th
e f
ixed
mo
du
les
an
d p
in c
oo
rdin
ate
s o
f all m
od
ule
s
Als
o n
eed
to
co
nstr
ain
t p
lacem
en
t s.t
. cen
ter-
of-
gra
vit
y (
the a
rea
weig
hte
d m
ean
of
all c
oo
rdin
ate
s)
co
rresp
on
ds t
o t
he c
en
ter
of
pla
cem
en
t re
gio
n:
Th
e v
ecto
r u
co
nta
ins g
eo
metr
ic c
en
ters
of
pla
cem
en
t re
gio
ns, th
e
matr
ix A
refl
ects
th
e a
ssig
nm
en
t o
f m
od
ule
s t
o p
lacem
en
t re
gio
ns
Φ=
+T
T(x
)0
.5x
Cx
dx
=A
xu
27
Ku
rt K
eu
tzer
Cost
=(x
1−
100)2
+(x
1−
x2)2
+(x
2−
200)2
�
�x 1
Cost
=2(x
1−
100)+
2(x
1−
x2)
�
�x 2
Cost
=−
2(x
1−
x2
)+
2(x
2−
200)
sett
ing
the
par
tial der
ivat
ives
= 0
we
solv
e fo
r th
e m
inim
um
Cost
:
Ax
+ B
= 0
=
04
−2
−2
4
x1
x2
+−200
−400
= 0
2−1
−1
2
x1
x2
+−100
−200
x1=
400/3
x
2=
500/3
x2
x1
x=
100
x=
200
Toy
Exam
ple
:
D. P
an
ρ ρρρ
Qu
ad
rati
c O
pti
miz
ati
on
Pro
ble
m
D
E
F
AB C
),
('
'ρ ρρρ
vu
= ===
M MMML LLLL LLLM MMM
M MMMM MMM
M MMMM MMM
M MMMM MMM
M MMMM MMM
M MMMM MMM
M MMMM MMM
M MMMM MMM
*0
*0
*0
00
0*
**
')
(
ρ ρρρρ ρρρl
A
GF
ED
CB
A
�L
inearl
y c
on
str
ain
ed
qu
ad
rati
c p
rog
ram
min
g p
rob
lem
)(
{m
inT
T
Rx
x }
dx
Cx
xm
+ +++= ===
Φ ΦΦΦ∈ ∈∈∈
ρ ρρρ)
,(
ρ ρρρv
u
s.t.
)(
)(
ll
ux
A= ===
Wir
e-l
en
gth
fo
r m
ovab
le m
od
ule
s
Ac
co
un
ts f
or
fix
ed
mo
du
les
Cen
ter-
of-
gra
vit
y c
on
str
ain
ts
Pro
ble
m i
s c
om
pu
tati
on
all
y t
racta
ble
, an
d w
ell b
eh
aved
Co
mm
erc
ial
so
lvers
avail
ab
le:
mo
ste
k
29
Ku
rt K
eu
tzer
Wh
at
do
I n
eed
to
kn
ow
ab
ou
t Q
P?
•G
et
an
in
tuit
ive s
en
se o
f w
hat
qu
ad
rati
c p
rog
ram
min
g is t
ryin
g
to s
olv
e
�C
reate
a g
oo
d p
lacem
en
t o
f th
e n
etl
ist
by p
lacin
g c
ell
s s
o
as t
o m
inim
ize t
he s
qu
are
s o
f th
e w
irele
ng
ths
in t
he n
etl
ist
•U
nd
ers
tan
d s
tren
gth
s a
nd
weakn
es
ses o
f th
e f
orm
ula
tio
n
�S
tren
gth
-o
pti
mal
so
luti
on
!
�W
eakn
ess –
�q
uad
rati
c v
s.
lin
ear
wir
ele
ng
ths
�N
eve
r kn
ow
s (
or
op
tim
izes)
the l
en
gth
of
a p
ath
-o
nly
th
e i
nd
ivid
ual 2-p
in n
ets
•D
on
’t n
eed
to
un
ders
tan
d t
he u
nd
erl
yin
g m
ath
em
ati
cs
Glo
bal O
pti
miz
ati
on
Usin
g Q
uad
rati
c
Pla
cem
en
t
Qu
ad
rati
c p
lacem
en
t clu
mp
s c
ells in
cen
ter
Part
itio
nin
g d
ivid
es c
ells in
to t
wo
reg
ion
s
�P
lacem
en
t re
gio
n is a
lso
div
ided
in
to t
wo
reg
ion
s
New
cen
ter-
of-
gra
vit
y c
on
str
ain
ts a
re a
dd
ed
to
th
e
co
nstr
ain
t m
atr
ix t
o b
e u
sed
on
th
e n
ext
level o
f g
lob
al
op
tim
izati
on
�G
lob
al co
nn
ecti
vit
y is s
till c
on
serv
ed
Sett
ing
up
Glo
bal O
pti
miz
ati
on
Layo
ut
Aft
er
Glo
bal O
pti
miz
ati
on
A. K
ah
ng
Part
itio
nin
g
34
Ku
rt K
eu
tzer
Part
itio
nin
g
In G
OR
DIA
N, p
art
itio
nin
g is u
sed
to
co
nstr
ain
t th
e m
ovem
en
t
of
mo
du
les r
ath
er
than
red
uce p
rob
lem
siz
e
By p
erf
orm
ing
part
itio
nin
g, w
e c
an
ite
rati
vely
im
po
se a
new
set
of
co
nstr
ain
ts o
n t
he g
lob
al o
pti
miz
ati
on
pro
ble
m
�A
ssig
n m
od
ule
s t
o a
part
icu
lar
blo
ck
Part
itio
nin
g is d
ete
rmin
ed
by
�R
esu
lts o
f g
lob
al p
lacem
en
t –
init
ial sta
rtin
g p
oin
t
�S
pati
al (x
,y)
dis
trib
uti
on
of
mo
du
les
�P
art
itio
nin
g c
ost
�W
an
t a m
in-c
ut
part
itio
n
35
Ku
rt K
eu
tzer
Part
itio
nin
g d
ue t
o G
lob
al O
pti
miz
ati
on
So
rt t
he m
od
ule
s b
y t
heir
x c
oo
rdin
ate
(fo
r a v
ert
ical cu
t)
Ch
oo
se a
cu
t lin
e s
uch
th
at
are
a is a
pp
roxim
ate
ly e
qu
al
betw
een
tw
o s
ides o
f cu
t
∈ ∈∈∈
∑ ∑∑∑∑ ∑∑∑
∈ ∈∈∈∈ ∈∈∈
≈ ≈≈≈= ===
∈ ∈∈∈≤ ≤≤≤→ →→→
Mu
uM
uu
pp
uu
pp
p
FF
Mu
Mu
xx
MM
M
pp
α ααα0.5
'',
',
''
'''
'''
'''
∈ ∈∈∈
Part
itio
nin
g Im
pro
vem
en
t -
I
∑ ∑∑∑
∑ ∑∑∑∑ ∑∑∑
∈ ∈∈∈
∈ ∈∈∈∈ ∈∈∈
= ===
≈ ≈≈≈= ===
∈ ∈∈∈≤ ≤≤≤→ →→→
Nc
v
v
Mu
uM
uu
pp
uu
pp
p
C
FF
Mu
Mu
xx
MM
M
pp
wα ααα
α ααα
)(
:cu
t valu
e
0.5
'',
',
p
''
'''
'''
'''
0.0
0
.25
0.5
0.7
5
1
.0
0
40
30
20
10
Cp(α ααα
)
•T
he c
ost
of
init
ial p
art
itio
n m
ay b
e t
oo
hig
h
•C
an
ch
an
ge p
osit
ion
of
the c
ut
to r
ed
uce t
he c
ost
•P
lot
the c
ost
fun
cti
on
, ch
oo
se “
best”
po
sit
ion
Layo
ut
aft
er
Min
-cu
t
No
w g
lob
al p
lacem
en
t p
rob
lem
will b
e s
olv
ed
ag
ain
w
ith
tw
o a
dd
itio
nal cen
ter_
of_
gra
vit
y c
on
str
ain
ts
Ad
din
g P
osit
ion
ing
Co
nstr
ain
ts
•P
art
itio
nin
g g
ives u
s t
wo
n
ew
“cen
ter
of
gra
vit
y”
co
nstr
ain
ts
•S
imp
ly u
pd
ate
co
nstr
ain
t m
atr
ix
•S
till a
sin
gle
glo
bal
op
tim
izati
on
pro
ble
m
•P
art
itio
nin
g is n
ot
“ab
so
lute
”•
mo
du
les c
an
mig
rate
b
ack d
uri
ng
op
tim
izati
on
•m
ay n
eed
to
re-p
art
itio
n
Co
nti
nu
e t
o Ite
rate
GO
RD
IAN
Pro
ced
ure
Go
rdia
nl:
=1;
glo
bal-
op
tim
ize(l
);
wh
ile(∃ ∃∃∃
|Ml|>
k)
for
each
ρ ρρρ∈ ∈∈∈
R(l
)p
art
itio
n(ρ ρρρ
,ρ ρρρ’,
ρ ρρρ”);
en
dfo
rl:
=l+
1;
setu
p-c
on
str
ain
ts(l
);g
lob
al-
op
tim
ize(l
);/*
extr
as
rep
art
itio
n(l
); *
/en
dw
hile
fin
al-
pla
cem
en
t(l)
;en
dp
roced
ure
41
Ku
rt K
eu
tzer
Fir
st
Itera
tio
n
A. K
ah
ng
42
Ku
rt K
eu
tzer
Seco
nd
Ite
rati
on
A. K
ah
ng
43
Ku
rt K
eu
tzer
Th
ird
Ite
rati
on
A. K
ah
ng
44
Ku
rt K
eu
tzer
Fo
urt
h Ite
rati
on
A. K
ah
ng
45
Ku
rt K
eu
tzer
GO
RD
IAN
(q
uad
rati
c +
part
itio
nin
g)
Par
titi
on
and R
epla
ce
Init
ial
Pla
cem
ent
A. K
ah
ng
46
Ku
rt K
eu
tzer
An
oth
er
Se
rie
s o
f G
ord
ian
(a)
Glo
ba
l p
lac
em
en
t w
ith
1 r
eg
ion
(b)
Glo
ba
l p
lac
em
en
t w
ith
4 r
eg
ion
(c)
Fin
al
pla
cem
en
ts
D. P
an
–U
of
Texas
Fin
al P
lacem
en
t
48
Ku
rt K
eu
tzer
Fin
al P
lacem
en
t -
1
Earl
ier
ste
ps h
ave b
roken
do
wn
th
e p
rob
lem
in
to a
man
ag
eab
le
nu
mb
er
of
ob
jects
Tw
o a
pp
roach
es:
�F
inal p
lacem
en
t fo
r sta
nd
ard
cells/g
ate
arr
ay –
row
assig
nm
en
t
�F
inal p
lacem
en
t fo
r la
rge, ir
reg
ula
rly s
ized
macro
-blo
cks –
slicin
g –
loo
k o
ver
in “
Extr
a”
slid
es a
t th
e e
nd
Fin
al P
lacem
en
t –
Sta
nd
ard
Cell D
esig
ns
Th
is p
rocess co
nti
nu
es u
nti
l th
ere
are
o
nly
a
few
cells in
each
gro
up
( ≈ ≈≈≈
6 )
each
gro
up
h
as ≤ ≤≤≤
6cells
gro
up
: sm
allest
part
itio
n
As
sig
n
cells
in
each
g
rou
p c
lose t
og
eth
er
in
the s
am
e r
ow
or
nearl
y
in a
dja
cen
t ro
ws
A.
E.
Du
nlo
p, B
. W
. K
ern
igh
an
, A
pro
ced
ure
fo
r p
lacem
en
t o
f s
tan
dard
-cell V
LS
I cir
cu
its, IE
EE
Tra
ns. o
n C
AD
, V
ol. C
AD
-4,
Jan
, 1
985,
pp
. 92-
98
50
Ku
rt K
eu
tzer
Sta
nd
ard
Cell L
ayo
ut
Fin
al P
lacem
en
t –
Cre
ati
ng
Ro
ws
11
11,2
1,2
1,2
1,2
22
2,3
2,3
2,3
2,3
33
3
3,4
3,4
3,4
3,4
44
44
5
55
5
5
5
4,5
4,5
a f
ou
r-ro
w
sta
nd
ard
cell
d
esig
n
Part
itio
nin
g o
f cir
cu
it in
to 32 g
rou
ps.
Each
g
rou
p is
eit
her
assig
ned
to
a sin
gle
ro
w o
r d
ivid
ed
in
to 2
ro
ws
Fin
al P
lacem
en
t –
Cre
ati
ng
Ro
ws
�P
art
itio
nin
g m
ust
be d
on
e b
read
th-f
irst
no
t d
ep
th f
irst
Cre
ati
ng
Ro
ws
C1
C2
C3
Ro
w 1
Ro
w 2
Ro
w 3
Ro
w 4
cells in
C1→ →→→
row
1
cells in
C3→ →→→
row
1
cells in
C2
C2
α ααα β βββ
α ααα+
β βββ=
1R
ow
1R
ow
2
Ch
oo
se α ααα
an
d β
β
β
β
pre
fera
bly
to
bala
nce r
ow
to
b
ala
nce r
ow
len
gth
(D
uri
ng
re-a
rran
gem
en
t )
53
Ku
rt K
eu
tzer
Fin
al P
lacem
en
t
Earl
ier
ste
ps h
ave b
roken
do
wn
th
e p
rob
lem
in
to a
man
ag
eab
le
nu
mb
er
of
ob
jects
Tw
o a
pp
roach
es:
�F
inal p
lacem
en
t fo
r sta
nd
ard
cells –
row
assig
nm
en
t
�F
inal p
lacem
en
t fo
r la
rge, ir
reg
ula
rly s
ized
macro
-blo
cks –
slicin
g –
see E
xtr
as
Gen
era
tin
g F
inal P
lacem
en
t
Exam
ple
Fin
al P
lacem
en
t
Su
mm
ary
of
sta
tus o
n p
lacem
en
t
Mo
st
pla
ce a
nd
ro
ute
to
ols
use
a s
eri
es o
f ``
half
-tru
ths’’
reg
ard
ing
tim
ing
mo
de
lin
g a
nd
dela
y c
on
str
ain
ts
No
t u
nch
ara
cte
risti
call
y,
the G
ord
ian
ap
pro
ach
uses t
hre
e d
iffe
ren
t ap
pro
ac
hes
to a
ttack t
he
pro
ble
m -
eac
h a
pp
roach
makes a
nu
mb
er
of
dif
fere
nt
sim
pli
ficati
on
s
Cu
rren
t g
ate
-le
vel p
lacem
en
t to
ols
(e.g
. A
po
llo
) are
ab
le t
o p
lac
e a
nd
ro
ute
hu
nd
red
s o
f th
ou
san
ds � ���
mil
lio
ns o
f ce
lls
fla
t (w
ith
ou
t h
iera
rch
y)
Desp
ite
th
e l
ack
of
dir
ec
t co
rre
lati
on
be
tween
th
e in
tern
al
tim
ing
mo
de
l an
d t
he
actu
al
inte
gra
ted
cir
cu
it t
imin
g -
tim
ing
clo
su
re b
etw
een
syn
thesis
an
d
ph
ys
ical
desig
n is i
mp
rovin
g b
ut
sti
ll a
pro
ble
m
Pri
ncip
all
y d
ue t
o:
•B
ett
er
de
lay e
sti
ma
tio
n i
n s
yn
thesis
•B
ett
er
de
lay c
alc
ula
tio
n i
n p
hys
ical
desig
n
•A
uto
ma
ted
bu
ffer
insert
ion
in
ro
uti
ng
57
Ku
rt K
eu
tzer
To
day’s
hig
h-p
erf
log
ical/p
hysic
al fl
ow
1)
op
tim
ize u
sin
g
esti
mate
d o
r
extr
acte
d
cap
acit
an
ces
2)
re-p
lace a
nd
re-
rou
te
3)i
f d
esig
n f
ails t
o
meet
co
nstr
ain
ts
du
e t
o p
oo
r
esti
mati
on
-re
peat
1 +
2-
netl
ist
Lib
rary
user
co
nstr
ain
ts
layo
ut
RC
extr
acti
on
dela
ym
od
el
gen
era
tor
rou
tin
g
tech
file
s
pla
cem
en
t
log
ico
pti
miz
ati
on
/ti
min
g v
eri
f
SD
Fcell/w
ire
dela
ys
58
Ku
rt K
eu
tzer
To
p-d
ow
n p
rob
lem
s in
th
e f
low
netl
ist
Lib
rary
user
co
nstr
ain
ts
layo
ut
RC
extr
acti
on
dela
ym
od
el
gen
era
tor
rou
tin
g
tech
file
s
pla
cem
en
t
log
ico
pti
miz
ati
on
/ti
min
g v
eri
f
SD
Fcell/w
ire
dela
ys
init
ial ca
pacit
an
ce
esti
ma
tes i
nacc
ura
te
inab
ilit
y t
o t
ake t
op
-d
ow
n t
imin
g
co
nstr
ain
ts
inaccu
rate
in
tern
al
tim
ing
mo
del
59
Ku
rt K
eu
tzer
Itera
tio
n p
rob
lem
s in
th
e f
low
netl
ist
Lib
rary
user
co
nstr
ain
ts
layo
ut
RC
extr
acti
on
dela
ym
od
el
gen
era
tor
rou
tin
g
tech
file
s
pla
cem
en
t
log
ico
pti
miz
ati
on
/ti
min
g v
eri
f
SD
Fcell/w
ire
dela
ys
up
da
ted
cap
acit
an
ces
cau
se s
ign
ific
an
t ch
an
ges i
n
op
tim
izati
on
lim
ited
-in
cre
men
tal
cap
ab
ilit
y
resu
ltin
g i
tera
tio
n m
ay
no
t b
rin
g c
loser
to
co
nverg
en
ce
Researc
h o
pp
ort
un
itie
s in
pla
cem
en
t
•B
ett
er
dela
y e
sti
mati
on
an
d m
od
elin
g in
syn
thesis
•B
ett
er
dela
y c
alc
ula
tio
n in
ph
ysic
al d
esig
n
•A
uto
mate
d r
esyn
thesis
du
rin
g p
lacem
en
t
•A
uto
mate
d s
izin
g a
nd
bu
ffer
insert
ion
in
ro
uti
ng
61
Ku
rt K
eu
tzer
Inte
gra
tin
g S
yn
thesis
an
d P
lacem
en
t
resi
zing
buff
erin
g
clonin
gre
stru
cturi
ng
Sta
n C
how
Am
mocore
Andre
w B
. K
ahng U
CS
DM
ajid
Sarr
afz
adeh U
CLA
62
Ku
rt K
eu
tzer
Oth
er
Ele
men
ts o
f In
du
str
ial F
low
Def
init
ion
s:•C
ell:
a c
ircu
it c
om
po
nen
t to
be
pla
ced
o
n t
he
chip
are
a.
In p
lace
men
t, t
he
fun
ctio
na
lity
of
the c
om
po
nen
t is
ig
no
red
.•N
et:
spec
ifyin
g a
su
bse
t o
f te
rmin
als
, t
o
con
nec
t se
vera
l ce
lls.
•Net
list
: a
set
of
net
s w
hic
h c
on
tain
s th
e co
nn
ecti
vit
y in
form
ati
on
of
the c
ircu
it.
Glo
ba
l P
lacem
en
t
Deta
il P
lacem
en
t
Clo
ck T
ree S
yn
thes
is
an
d R
ou
tin
g
Glo
ba
l R
ou
tin
g
Deta
il R
ou
tin
g
Po
wer/
Gro
un
d
Str
ipe
s, R
ing
s R
ou
tin
g
Extr
ac
tio
n a
nd
D
ela
y C
alc
. T
imin
g
Veri
fic
ati
on
IO P
ad
Pla
cem
en
t
A. K
ah
ng
63
Ku
rt K
eu
tzer
Extr
as
•Wo
rked
ou
t exa
mp
le f
or
qu
ad
rati
c p
rog
ram
min
g –
D. P
an
, U
of
Texas
•Bri
ef
su
rve
y o
f ap
pro
ach
es
•Mo
re o
n q
uad
rati
c p
lace
men
t
•Usin
g s
lic
ing
to
han
dle
macro
-blo
cks
64
Ku
rt K
eu
tzer
Cost
=(x
1−
100)2
+(x
1−
x2)2
+(x
2−
200)2
�
�x 1
Cost
=2(x
1−
100)+
2(x
1−
x2)
�
�x 2
Cost
=−
2(x
1−
x2
)+
2(x
2−
200)
sett
ing
the
par
tial der
ivat
ives
= 0
we
solv
e fo
r th
e m
inim
um
Cost
:
Ax
+ B
= 0
=
04
−2
−2
4
x1
x2
+−200
−400
= 0
2−1
−1
2
x1
x2
+−100
−200
x1=
400/3
x
2=
500/3
x2
x1
x=
100
x=
200
Toy
Exam
ple
:
D. P
an
65
Ku
rt K
eu
tzer
settin
g th
e pa
rtia
l der
ivat
ives
= 0
we
solv
e fo
r th
e m
inim
um C
ost:
Ax
+ B
= 0
=
04
−2
−24
x 1 x 2+
−200
−400
= 0
2−1
−12
x 1 x 2+
−100
−200
x1=40
0/3
x2=
500/
3
x2
x1
x=
100
x=
200
Inte
rpre
tation o
f m
atr
ices A
and B
:
The d
iagonal valu
es A
[i,i] corr
esp
ond to t
he n
um
ber
of connections to x
i
The o
ff d
iagonal valu
es A
[i,j] are
-1 if obje
ct i is
connecte
d to o
bje
ct j, 0
oth
erw
ise
The v
alu
es B
[i] corr
espond to t
he s
um
of
the locations o
f fixed o
bje
cts
connecte
d to o
bje
ct i
Exam
ple
:
D. P
an
66
Ku
rt K
eu
tzer
Tra
dit
ion
al A
pp
roach
es
•Q
uad
rati
c P
lacem
en
t
•S
imu
late
d A
nn
ealin
g
•B
i-P
art
itio
nin
g
•Q
uad
risecti
on
•F
orc
e D
irecte
d P
lacem
en
t
•H
yb
rid
Overv
iew
of G
ord
ian P
ackag
e
Pro
ced
ure
Go
rdia
nl:
=1;
glo
bal-
op
tim
ize(l
);
wh
ile(∃ ∃∃∃
|Ml|>
k)
for
each
ρ ρρρ∈ ∈∈∈
R(l
)p
art
itio
n(ρ ρρρ
,ρ ρρρ’,
ρ ρρρ”);
en
dfo
rl:
=l+
1;
setu
p-c
on
str
ain
ts(l
);g
lob
al-
op
tim
ize(l
);re
part
itio
n(l
);en
dw
hile
fin
al-
pla
cem
en
t(l)
;en
dp
roced
ure
GO
RD
IAN
wit
h r
ep
art
itio
nin
g p
roced
ure
Cost F
unction
TT
XX
Χ ΧΧΧd
Cx
+ +++= ===
)(
φ φφφ
YY
XX
Χ ΧΧΧY
dC
dC
yx
T yT
T xT
+ ++++ +++
+ +++= ===
),
(φ φφφ
uv
ξ ξξξ
1∑ ∑∑∑
Lv
Nv
v= ===
∈ ∈∈∈2
wφ φφφO
ve
rall
ob
jecti
ve
�T
he c
en
ter
of
gra
vit
y c
on
str
ain
ts
At
level l, c
hip
is d
ivid
ed
in
to q
( ≤ ≤≤≤2
l )
reg
ion
s
Fo
r re
gio
n p
. th
e c
en
ter
co
ord
inate
s:
(up,
vp)
(Mp
: set
of
mo
du
les in
reg
ion
p)
Matr
ix f
orm
fo
r all r
eg
ion
s
Glo
bal P
lacem
ent and C
onstr
ain
ts
∑ ∑∑∑∑ ∑∑∑
∈ ∈∈∈∈ ∈∈∈
= ===
pp
Mu
up
uM
uu
Fu
xF
:s
con
stra
int
0
,)(
)(
∑ ∈ ∈∈∈= ===
= ===p
Mi
ii
pu
ll
FF
au
XA
If i
∈ ∈∈∈M
p
oth
erw
ise
ρ ρρρ
Pro
ble
m F
orm
ula
tion
D
E
F
AB C
),
('
'ρ ρρρ
vu
= ===
M MMML LLLL LLLM MMM
M MMMM MMM
M MMMM MMM
M MMMM MMM
M MMMM MMM
M MMMM MMM
M MMMM MMM
M MMMM MMM
*0
*0
*0
00
0*
**
')
(
ρ ρρρρ ρρρl
A
GF
ED
CB
A
�L
inearl
y c
on
str
ain
ed
qu
ad
rati
c p
rog
ram
min
g p
rob
lem
}s.
t.)
({
min
:L
QP
)(
)(
ll
TT
Rx
uX
AX
dX
CX
xm
= ===+ +++
= ===Φ ΦΦΦ
∈ ∈∈∈
ρ ρρρ)
,(
ρ ρρρv
u
Solu
tion M
eth
od
[ [[[] ]]]
[ [[[] ]]]
0
11
11
)(
0
,
xZ
Xu
DX
I
BD
X
uD
BX
DX
uXX
BD
BD
A
i
id
id
qm
qm
q
+ +++= ===
+ +++
− −−−= ===
+ +++− −−−
= ===
= ===
= ===
− −−−− −−−
− −−−− −−−
− −−−× ×××
× ×××× ×××
Xd
dep
en
den
t vari
ab
les
Xi
ind
ep
en
den
t vari
ab
les
un
co
nstr
ain
ed
qu
ad
rati
c p
rog
ram
min
g p
rob
lem
)(C
})
({
min
:U
QP
0
Td
CX
XC
CZ
XZ
Xx
iT
TT i
iR
xq
mi
+ +++= ===
+ +++= ===
− −−−∈ ∈∈∈
ψ ψψψ
So
lved
by c
on
jug
ate
--g
rad
ien
t m
eth
od
p
Term
ina
l P
ropag
atio
n
L1
L2
R1
R2
Pre
fer
to h
ave
all o
f th
em
in
R1
RL
1
L2
net
s
�W
e s
ho
uld
use t
he f
act
that
s is in
L1!
Fic
titi
ou
s c
ell
o
f n
et
s
L1
L2
R1
R2
As
su
min
g
locate
d a
t cen
ter
L1
L2
R1
R2
pL
1
L2
R1
R2
p
hig
her
co
st
low
er
co
st
p w
ill
sta
y i
n R
1fo
r th
e r
est
of
part
itio
nin
g
73
Ku
rt K
eu
tzer
Macro
Pla
cem
en
t b
y S
licin
g
Co
nstr
ain
ing
Pla
cem
en
t o
f M
acro
s
Insert
yo
ur
log
ic h
ere
Glo
bal o
pti
miz
ati
on
an
d p
art
itio
nin
g a
ssig
ns <
=k c
ell
s t
o e
ach
ph
ysic
al
reg
ion
Th
ere
are
dif
fere
nt
ways o
f p
lacin
g c
ell
s in
each
reg
ion
Wan
t to
ch
oo
se p
lacem
en
ts t
hat
min
imiz
e t
ota
l ch
ip a
rea
Macro
-blo
cks:
Exh
au
sti
ve S
licin
g
Op
tim
izati
on
L. van
Gin
nekin
, R
. H
. O
tten
, O
pti
mal S
licin
g o
f P
oin
t P
lacem
en
ts, E
DA
C,
1990, p
p. 322-3
26
Sli
cin
g v
s.
No
n-S
lic
ing
Flo
orp
lan
s
Slicin
g F
loo
rpla
n(s
lic
ing
str
uc
ture
): A
recta
ng
le d
issec
tio
n w
hic
h c
an
be
ob
tain
ed
by r
ec
urs
ively
dis
sec
tin
g t
he t
he b
ase
recta
ng
le in
to
sm
all
er
rec
tan
gle
s b
y v
ert
ical
an
d h
ori
zo
nta
l sli
cin
g l
ines
.
Slicin
g s
tru
ctu
res a
re g
oo
d f
or
rou
tin
g
Sli
cing F
loorp
lan
Non-S
lici
ng F
loorp
lan(w
hee
l)
Slicin
g T
ree
A
BC
D
E F G
D
A
C
B
GF
E
H
VH
HH
V
Slicin
g s
tru
ctu
re is d
escri
bed
by a
slicin
g t
ree
Th
e S
hap
e A
lgo
rith
m
Inp
ut:
S
licin
g T
ree
Sh
ap
e C
on
str
ain
ts f
or
mo
du
les
Co
st
Fu
ncti
on
(n
on
-decre
asin
g in
w, h
)
Ou
tpu
t:S
hap
es/Im
ple
men
tati
on
fo
r each
mo
du
le
Alg
ori
thm
•C
om
po
se s
hap
e c
on
str
ain
ts b
ott
om
-up
in
th
e s
licin
g t
ree
•A
pp
ly c
ost
fun
cti
on
to
co
mp
ute
bo
un
dary
po
int
on
sh
ap
e
co
nstr
ain
t o
f b
ase b
lock (
roo
t)
•P
rop
ag
ate
bo
un
dary
po
int
top
-do
wn
in
slicin
g t
ree t
o o
bta
in
imp
lem
en
tati
on
fo
r each
mo
du
le
Sh
ap
e C
on
str
ain
ts
Giv
en
a r
ec
tan
gu
lar
mo
du
le,
the s
hap
e c
on
str
ain
t re
lati
on
Ris
th
e s
et
of
y-x
pair
s s
o t
ha
t a
recta
ng
le w
ith
wid
th e
qu
al
to x
an
d h
eig
ht
eq
ua
l to
y c
on
tain
s a
t le
ast
a s
hap
e/o
rien
tati
on
realizati
on
of
the m
od
ule
.
3
1
3
1
12
34
12
34
2 134
2 134
80
Ku
rt K
eu
tzer
Co
mp
osin
g S
hap
e C
on
str
ain
ts
3 b
y 1
4 b
y 1
1
2
3
4 5
6
7
5 4 3 2 1
1
2
3
4 5
6
7
5 4 3 2 1
1
2
3
4 5
6
7
7 6 5 4 3 2 1
7 b
y 1
Slicin
g T
ree
A
BC
D
E F G
D
A
C
B
GF
E
H
VH
HH
V
Tra
vers
e t
ree b
ott
om
-up
deri
vin
g c
om
po
sed
sh
ap
e f
un
cti
on
s
Th
e S
hap
e A
lgo
rith
m
Inp
ut:
S
licin
g T
ree
Sh
ap
e C
on
str
ain
ts f
or
mo
du
les
Co
st
Fu
ncti
on
(n
on
-decre
asin
g in
w, h
)
Ou
tpu
t:S
hap
es/Im
ple
men
tati
on
fo
r each
mo
du
le
Alg
ori
thm
•C
om
po
se s
hap
e c
on
str
ain
ts b
ott
om
-up
in
th
e s
licin
g t
ree
•A
pp
ly c
ost
fun
cti
on
to
co
mp
ute
bo
un
dary
po
int
on
sh
ap
e
co
nstr
ain
t o
f b
ase b
lock (
roo
t)
•P
rop
ag
ate
bo
un
dary
po
int
top
-do
wn
in
slicin
g t
ree t
o o
bta
in
imp
lem
en
tati
on
fo
r each
mo
du
le
Co
nstr
ain
ing
Pla
cem
en
t o
f M
acro
s
Insert
yo
ur
log
ic h
ere
30 X
50
At
top
level can
have c
om
ple
x s
hap
e f
un
cti
on
s
Ap
ply
a c
ost
fun
cti
on
an
d s
ele
ct
low
est
sco
re p
oin
t, e
.g.
�co
st(
w,h
) =
wh
�C
ost(
w,h
) =
2(w
+h
)
Slicin
g T
ree
D
A
C
B
GF
E
H
VH
HH
V30 X
50
Th
e S
hap
e A
lgo
rith
m
Inp
ut:
S
licin
g T
ree
Sh
ap
e C
on
str
ain
ts f
or
mo
du
les
Co
st
Fu
ncti
on
(n
on
-decre
asin
g in
w, h
)
Ou
tpu
t:S
hap
es/Im
ple
men
tati
on
fo
r each
mo
du
le
Alg
ori
thm
•C
om
po
se s
hap
e c
on
str
ain
ts b
ott
om
-up
in
th
e s
licin
g t
ree
•A
pp
ly c
ost
fun
cti
on
to
co
mp
ute
bo
un
dary
po
int
on
sh
ap
e
co
nstr
ain
t o
f b
ase b
lock (
roo
t)
•P
rop
ag
ate
bo
un
dary
po
int
top
-do
wn
in
slicin
g t
ree t
o o
bta
in
imp
lem
en
tati
on
fo
r each
mo
du
le
Slicin
g T
ree
A
BC
D
E F G
D
A
C
B
GF
E
H
VH
HH
V
Pro
pag
ate
op
tim
al ch
oic
es t
op
-do
wn
to
all t
he leaves
Min
-Cut
Based P
lace
me
nt
(Cont’d
)
A
BC
D
E F G
