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Par
titi
on
ing
fo
r
Ph
ysic
al D
esig
n
Pro
f. A
. R. N
ewto
n
Pro
f. K
. Keu
tzer
Mic
hae
l Ors
han
sky
EE
CS
Un
iver
sity
of
Cal
ifo
rnia
Ber
kele
y, C
A
Wit
h a
dd
itio
nal
mat
eria
l fro
m A
nd
rew
B. K
ahn
g, U
CS
D, M
. Sar
rafz
aW
ith
ad
dit
ion
al m
ater
ial f
rom
An
dre
w B
. Kah
ng
, UC
SD
, M. S
arra
fza
deh
, UC
LA
deh
, UC
LA
EE
244
2
Let
’s t
ake
a st
ep b
ack
to t
he
1980
’s
Eff
ort
(ED
A t
oo
ls e
ffo
rt)
Resu
lts
(Desig
n P
rod
ucti
vit
y)
a b
s
q
0 1
d
clk
1978
1978
1978
1978
1985
1985
1985
1985
1992
1992
1992
1992
1999
1999
1999
1999
Tra
nsis
tor
en
try -
Calm
a, C
om
pu
terv
isio
n
Sch
em
ati
c E
ntr
y -
Dais
y, M
en
tor,
Valid
Syn
thesis
-C
ad
en
ce, S
yn
op
sys
Wh
at’
s n
ext?
McK
inse
y S
-Cu
rve
EE
244
3
Sch
emat
ic 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
EE
244
4
Bas
ic P
hys
ical
Des
ign
Pro
ble
mB
asic
Ph
ysic
al D
esig
n P
rob
lem
�W
hat
pro
ble
ms
nee
d t
o b
e so
lved
in
ph
ysic
al d
esig
n?
�W
hat
pro
ble
ms
nee
d t
o b
e so
lved
in
ph
ysic
al d
esig
n?
Sch
emat
ic
Lay
ou
t
EE
244
5
Bas
ic P
hys
ical
Des
ign
Pro
ble
mB
asic
Ph
ysic
al D
esig
n P
rob
lem
�W
hat
pro
ble
ms
need
to
be
so
lved
in
ph
ysic
al
desig
n?
�P
lan
ari
ze
gra
ph
� ���p
lace g
ate
s/c
ells
�R
ou
te w
ires t
o c
on
nect
cell
s�
Ro
ute
clo
ck
�R
ou
te p
ow
er
an
d g
rou
nd
�B
on
d I
/O’s
to
I/O
pad
s
�W
hat
pro
ble
ms
need
to
be
so
lved
in
ph
ysic
al
desig
n?
�P
lan
ari
ze
gra
ph
� ���p
lace g
ate
s/c
ells
�R
ou
te w
ires t
o c
on
nect
cell
s�
Ro
ute
clo
ck
�R
ou
te p
ow
er
an
d g
rou
nd
�B
on
d I
/O’s
to
I/O
pad
s
Sch
emat
ic
Lay
ou
t
EE
244
6
Ph
ysic
al D
esig
n:
Ove
rall
Flo
wR
ead
Net
list
Init
ial P
lace
men
t
Pla
cem
ent
Imp
rove
men
t
Co
st E
stim
atio
n
Ro
uti
ng
Reg
ion
Def
init
ion
Glo
bal
Ro
uti
ng
Inp
ut
Pla
cem
ent
Ro
uti
ng
Ou
tpu
tC
om
pac
tio
n/c
lean
-up
Ro
uti
ng
Reg
ion
Ord
erin
g
Det
aile
d R
ou
tin
g
Co
st E
stim
atio
n
Ro
uti
ng
Imp
rove
men
t
Wri
te L
ayo
ut
Dat
abas
e
Flo
orp
lan
nin
gF
loo
rpla
nn
ing
EE
244
7
Fo
rmu
lati
on
of
the
Pla
cem
ent
Pro
ble
mF
orm
ula
tio
n o
f th
e P
lace
men
t P
rob
lem
�G
iven
:
�A
net
list
of
cells
fro
m a
pre
-def
ined
sem
ico
nd
uct
or
libra
ry
�A
mat
hem
atic
al e
xpre
ssio
n o
f th
at n
etlis
t as
a v
erte
x-, e
dg
e-w
eig
hte
d
gra
ph
�C
on
stra
ints
on
pin
-lo
cati
on
s ex
pre
ssed
as
con
stra
ints
on
ver
tex
loca
tio
ns
/ asp
ect
rati
o t
hat
th
e p
lace
men
t n
eed
s to
fit
into
�O
ne
or
mo
re o
f th
e fo
llow
ing
: ch
ip-l
evel
tim
ing
co
nst
rain
ts, a
list
of
crit
ical
net
s, c
hip
-lev
el p
ow
er c
on
stra
ints
�F
ind
:
�C
ell/v
erte
x lo
cati
on
s to
min
imiz
e p
lace
men
t o
bje
ctiv
e su
bje
ct t
o
con
stra
ints
�T
ypic
al O
bje
ctiv
es:
�m
inim
al d
elay
(fa
stes
t cl
ock
cyc
le t
ime)
�m
inim
al a
rea
(lea
st d
ie a
rea/
cost
�m
inim
al p
ow
er (
stat
ic, d
ynam
ic)
�G
iven
:
�A
net
list
of
cells
fro
m a
pre
-def
ined
sem
ico
nd
uct
or
libra
ry
�A
mat
hem
atic
al e
xpre
ssio
n o
f th
at n
etlis
t as
a v
erte
x-, e
dg
e-w
eig
hte
d
gra
ph
�C
on
stra
ints
on
pin
-lo
cati
on
s ex
pre
ssed
as
con
stra
ints
on
ver
tex
loca
tio
ns
/ asp
ect
rati
o t
hat
th
e p
lace
men
t n
eed
s to
fit
into
�O
ne
or
mo
re o
f th
e fo
llow
ing
: ch
ip-l
evel
tim
ing
co
nst
rain
ts, a
list
of
crit
ical
net
s, c
hip
-lev
el p
ow
er c
on
stra
ints
�F
ind
:
�C
ell/v
erte
x lo
cati
on
s to
min
imiz
e p
lace
men
t o
bje
ctiv
e su
bje
ct t
o
con
stra
ints
�T
ypic
al O
bje
ctiv
es:
�m
inim
al d
elay
(fa
stes
t cl
ock
cyc
le t
ime)
�m
inim
al a
rea
(lea
st d
ie a
rea/
cost
�m
inim
al p
ow
er (
stat
ic, d
ynam
ic)
EE
244
8
Res
ult
s o
f P
lace
men
tR
esu
lts
of
Pla
cem
ent
A b
ad
pla
cem
ent
A g
oo
d p
lace
men
t
A. K
ah
ng
EE
244
9
Glo
bal
an
d D
etai
led
Pla
cem
ent
Glo
bal
an
d D
etai
led
Pla
cem
ent
Glo
bal
Pla
cem
ent
Det
ail
ed P
lace
men
t
In g
lob
al p
lace
men
t, w
e
dec
ide
the
app
roxim
ate
loca
tion
s fo
r ce
lls
by
pla
cin
g c
ells
in g
lob
al b
ins.
In d
etai
led
pla
cem
ent,
w
e
mak
e so
me
loca
l ad
just
men
t
to o
bta
in t
he
fin
al n
on-
ov
erla
pp
ing p
lace
men
t.
A. K
ah
ng
EE
244
10
P
lace
men
t F
oo
tpri
nts
:
Sta
ndard
Cell:
Data
Path
:
IP b
lock -
Flo
orp
lannin
g
A. K
ah
ng
EE
244
11
Co
re
Co
ntr
ol
IO
Reserv
ed
are
as
Mix
ed
Data
Path
&sea o
f g
ate
s:
P
lace
men
t F
oo
tpri
nts
:
A. K
ah
ng
EE
244
12
Peri
mete
r IO
Are
a IO
–b
all g
rid
arr
ay
P
lace
men
t F
oo
tpri
nts
:
A. K
ah
ng
EE
244
13
Ap
pro
ach
to
Pla
cem
ent:
GO
RD
IAN
1A
pp
roac
h t
o P
lace
men
t: G
OR
DIA
N 1
EE
244
14
GO
RD
IAN
(q
uad
rati
c +
par
titi
on
ing
)G
OR
DIA
N (
qu
adra
tic
+ p
arti
tio
nin
g)
Init
ial
Pla
cem
ent
A. K
ah
ng
EE
244
15
Ap
pro
ach
to
Pla
cem
ent
: G
OR
DIA
N 2
Ap
pro
ach
to
Pla
cem
ent
: G
OR
DIA
N 2
EE
244
16
Par
titi
on
in G
OR
DIA
NP
arti
tio
n in
GO
RD
IAN
Par
titi
on
and R
epla
ce
A. K
ah
ng
EE
244
17
GO
RD
IAN
(q
uad
rati
c +
par
titi
on
ing
)G
OR
DIA
N (
qu
adra
tic
+ p
arti
tio
nin
g)
Par
titi
on
and R
epla
ce
Init
ial
Pla
cem
ent
A. K
ah
ng
EE
244
18
Bas
ic Id
ea o
f P
arti
tio
nin
gB
asic
Idea
of
Par
titi
on
ing
�P
arti
tio
n d
esig
n in
to t
wo
(g
ener
ally
N)
equ
al s
ize
hal
ves
�M
inim
ize
wir
es (
net
s) w
ith
en
ds
in b
oth
hal
ves
�N
um
ber
of
wir
es c
ross
ing
is b
isec
tion
band
wid
th
�lo
wer
bw
= m
ore
loca
lity
�P
arti
tio
n d
esig
n in
to t
wo
(g
ener
ally
N)
equ
al s
ize
hal
ves
�M
inim
ize
wir
es (
net
s) w
ith
en
ds
in b
oth
hal
ves
�N
um
ber
of
wir
es c
ross
ing
is b
isec
tion
band
wid
th
�lo
wer
bw
= m
ore
loca
lity
N/2
N/2
cuts
ize
EE
244
19
Net
list
Par
titi
on
ing
: M
oti
vati
on
1N
etlis
t P
arti
tio
nin
g:
Mo
tiva
tio
n 1
�D
ivid
ing
a n
etl
ist
into
clu
ste
rs t
o
�R
ed
uc
e p
rob
lem
siz
e
�E
vo
lve t
ow
ard
a p
hysic
al
pla
cem
en
t
�A
ll t
op
-do
wn
pla
ce
me
nt
ap
pro
ac
he
s u
tili
ze
so
me
un
de
rlyin
g p
art
itio
nin
g t
ec
hn
iqu
e
�In
flu
en
ce
s t
he
fin
al
qu
ali
ty o
f
�P
lacem
en
t
�G
lob
al ro
uti
ng
�D
eta
iled
ro
uti
ng
�D
ivid
ing
a n
etl
ist
into
clu
ste
rs t
o
�R
ed
uc
e p
rob
lem
siz
e
�E
vo
lve t
ow
ard
a p
hysic
al
pla
cem
en
t
�A
ll t
op
-do
wn
pla
ce
me
nt
ap
pro
ac
he
s u
tili
ze
so
me
un
de
rlyin
g p
art
itio
nin
g t
ec
hn
iqu
e
�In
flu
en
ce
s t
he
fin
al
qu
ali
ty o
f
�P
lacem
en
t
�G
lob
al ro
uti
ng
�D
eta
iled
ro
uti
ng
EE
244
20
Net
list
Par
titi
on
ing
: M
oti
vati
on
2N
etlis
t P
arti
tio
nin
g:
Mo
tiva
tio
n 2
�B
ec
om
es
mo
re c
riti
ca
l w
ith
DS
M
�S
ys
tem
siz
e i
nc
rea
se
s�
Ne
ed
to
min
imiz
e d
esig
n c
ou
plin
g
�In
terc
on
ne
ct
do
min
ate
s c
hip
pe
rfo
rma
nc
e�
Ha
ve t
o m
inim
ize n
um
ber
of
blo
ck-t
o-b
lock
co
nn
ecti
on
s (
e.g
. g
lob
al b
uses)
�H
elp
s r
ed
uc
e c
hip
are
a�
Min
imiz
es len
gth
of
glo
bal w
ires
�B
ec
om
es
mo
re c
riti
ca
l w
ith
DS
M
�S
ys
tem
siz
e i
nc
rea
se
s�
Ne
ed
to
min
imiz
e d
esig
n c
ou
plin
g
�In
terc
on
ne
ct
do
min
ate
s c
hip
pe
rfo
rma
nc
e�
Ha
ve t
o m
inim
ize n
um
ber
of
blo
ck-t
o-b
lock
co
nn
ecti
on
s (
e.g
. g
lob
al b
uses)
�H
elp
s r
ed
uc
e c
hip
are
a�
Min
imiz
es len
gth
of
glo
bal w
ires
EE
244
21
Par
titi
on
ing
fo
r M
inim
um
Cu
t-S
et
(a)
Ori
gin
al P
arti
tio
n (
Ran
do
m)
(b)
Imp
rove
d P
arti
tio
n
EE
244
22
Gra
ph
s an
d H
yper
gra
ph
sG
rap
hs
and
Hyp
erg
rap
hs
� � � �A
cir
cu
it n
etl
ist
is a
hyp
erg
rap
h
� � � �A
cir
cu
it n
etl
ist
is a
hyp
erg
rap
h
=
=≡
A g
rap
h
V -
ve
rte
x s
et,
E -
ed
ge
se
t, a
bin
ary
re
lati
on
sh
ip o
n V
.
G(V
,E).
e(v
,v).
e2
.i
i1i2
i
In a
n u
nd
ire
cte
d g
rap
h,
the
ed
ge
se
t c
on
sis
ts o
f u
no
rde
red
pa
irs
of
ve
rtic
es
.
In a
hyp
erg
rap
h,
H a
hyp
ere
dg
e
co
nn
ec
ts a
n a
rbit
rary
su
bs
et
of
ve
rtic
es
,
e.g
. i
e2
.
(V,E
),e
≥
EE
244
23
Net
list
Par
titi
on
ing
Net
list
Par
titi
on
ing
A
F
E
D
C
B
G
A
F
E
D
C
B
G Fir
st p
rob
lem
tra
nsi
tio
n f
rom
mu
lti-
term
inal
to
tw
o t
erm
inal
ed
ges
EE
244
24
Ed
ge
Wei
gh
ts f
or
Mu
ltit
erm
inal
Net
sE
dg
e W
eig
hts
fo
r M
ult
iter
min
alN
ets
�E
dg
es r
epre
sen
t n
ets
in t
he
circ
uit
net
list
�E
ach
ed
ge
in t
he
hyp
erg
rap
hw
ill t
ypic
ally
be
giv
en a
wei
gh
t w
hic
h r
epre
sen
ts it
s
crit
ical
ity
(cf.
tim
ing
lect
ure
)
�T
hes
e w
eig
hts
will
be
use
d t
o “
dri
ve”
par
titi
on
ing
, pla
cem
ent,
an
d r
ou
tin
g
�B
ut
if w
e w
ant
to u
se a
gra
ph
str
uct
ure
, as
op
po
sed
to
a h
yper
gra
ph
, we
mu
st r
e-d
efin
e
the
edg
es a
nd
th
eir
wei
gh
ts
�E
dg
es r
epre
sen
t n
ets
in t
he
circ
uit
net
list
�E
ach
ed
ge
in t
he
hyp
erg
rap
hw
ill t
ypic
ally
be
giv
en a
wei
gh
t w
hic
h r
epre
sen
ts it
s
crit
ical
ity
(cf.
tim
ing
lect
ure
)
�T
hes
e w
eig
hts
will
be
use
d t
o “
dri
ve”
par
titi
on
ing
, pla
cem
ent,
an
d r
ou
tin
g
�B
ut
if w
e w
ant
to u
se a
gra
ph
str
uct
ure
, as
op
po
sed
to
a h
yper
gra
ph
, we
mu
st r
e-d
efin
e
the
edg
es a
nd
th
eir
wei
gh
ts
P1
P2
Pn
EE
244
25
Ed
ge
Wei
gh
ts f
or
Mu
ltit
erm
inal
Net
sE
dg
e W
eig
hts
fo
r M
ult
iter
min
alN
ets
�R
epla
ce e
ach
net
Siw
ith
its
com
ple
te g
rap
h.
�W
hat
wei
gh
t o
n e
ach
ed
ge?
�O
ne
app
roac
h –
assi
gn
wei
gh
t o
f 1
to e
ach
net
in t
he
new
gra
ph
�A
lter
nat
ive:
n-p
in n
et, w
=2/
(n-1
) h
as b
een
use
d, a
lso
w=
2/n
�“S
tan
dar
d”
mo
del
: f
or
n n
ets
in t
he
com
ple
te g
rap
h
w=
1/(n
-1)
�F
or
any
cut,
co
st >
= 1
�L
arg
e n
ets
are
less
like
ly t
o b
e cu
t
�L
ead
s to
hig
hly
su
b-o
pti
mal
par
titi
on
s
�P
rovi
des
an
up
per
bo
un
do
n t
he
cost
of
a cu
t in
th
e ac
tual
net
list
�H
ow
ab
ou
t a
low
er b
ou
nd
on
th
e cu
t co
st?
�R
epla
ce e
ach
net
Siw
ith
its
com
ple
te g
rap
h.
�W
hat
wei
gh
t o
n e
ach
ed
ge?
�O
ne
app
roac
h –
assi
gn
wei
gh
t o
f 1
to e
ach
net
in t
he
new
gra
ph
�A
lter
nat
ive:
n-p
in n
et, w
=2/
(n-1
) h
as b
een
use
d, a
lso
w=
2/n
�“S
tan
dar
d”
mo
del
: f
or
n n
ets
in t
he
com
ple
te g
rap
h
w=
1/(n
-1)
�F
or
any
cut,
co
st >
= 1
�L
arg
e n
ets
are
less
like
ly t
o b
e cu
t
�L
ead
s to
hig
hly
su
b-o
pti
mal
par
titi
on
s
�P
rovi
des
an
up
per
bo
un
do
n t
he
cost
of
a cu
t in
th
e ac
tual
net
list
�H
ow
ab
ou
t a
low
er b
ou
nd
on
th
e cu
t co
st?
P1
P2
Pn
P1
P2 P
n
EE
244
26
Ed
ge
Wei
gh
ts f
or
Mu
ltit
erm
inal
Net
sE
dg
e W
eig
hts
fo
r M
ult
iter
min
alN
ets
P1
P2
Pn
11/
2
1/2
1/2
11/
4
1/4
1/4
1/4
1/4
1/4
EE
244
27
An
oth
er W
eig
ht
Ass
ign
men
t fo
r L
ow
er B
ou
nd
ing
the
Net
Cu
tA
no
ther
Wei
gh
t A
ssig
nm
ent
for
Lo
wer
Bo
un
din
gth
e N
et C
ut
�W
ant
to f
ind
a w
eig
ht
assi
gn
men
t th
at a
lway
s u
nd
eres
tim
ates
net
cu
ts�
Giv
es a
low
er b
ou
nd
on
th
e co
st o
f th
e n
etlis
t cu
t
�In
tuit
ivel
y: c
ho
ose
wei
gh
t as
sig
nm
ent
s.t
max
co
st o
f a
net
cu
t in
a
gra
ph
is 1
.
�M
axim
um
co
st h
app
ens
wh
en n
od
es a
re d
ivid
ed e
qu
ally
bet
wee
n 2
p
arti
tio
ns
�T
he
nu
mb
er o
f cr
oss
ing
ed
ges
in t
hat
sit
uat
ion
(p
roo
f le
ft t
o t
he
read
er ☺ ☺☺☺
)
�(n
2-m
od
(n,2
))/4
Eac
h e
dg
e is
ass
ign
ed t
he
wei
gh
t o
f
w =
4/(
n2-m
od(n
,2))
Exam
ple
: fo
r n
=3,
w=
4/(9
-1)=
0.5
�W
ant
to f
ind
a w
eig
ht
assi
gn
men
t th
at a
lway
s u
nd
eres
tim
ates
net
cu
ts�
Giv
es a
low
er b
ou
nd
on
th
e co
st o
f th
e n
etlis
t cu
t
�In
tuit
ivel
y: c
ho
ose
wei
gh
t as
sig
nm
ent
s.t
max
co
st o
f a
net
cu
t in
a
gra
ph
is 1
.
�M
axim
um
co
st h
app
ens
wh
en n
od
es a
re d
ivid
ed e
qu
ally
bet
wee
n 2
p
arti
tio
ns
�T
he
nu
mb
er o
f cr
oss
ing
ed
ges
in t
hat
sit
uat
ion
(p
roo
f le
ft t
o t
he
read
er ☺ ☺☺☺
)
�(n
2-m
od
(n,2
))/4
Eac
h e
dg
e is
ass
ign
ed t
he
wei
gh
t o
f
w =
4/(
n2-m
od(n
,2))
Exam
ple
: fo
r n
=3,
w=
4/(9
-1)=
0.5
EE
244
28
Par
titi
on
ing
�G
iven
a g
rap
h, G
, wit
h n
no
des
wit
h s
izes
(w
eig
hts
) w
:
wit
h c
ost
s o
n it
s ed
ges
, par
titi
on
th
e n
od
es o
f G
into
k, s
ub
sets
, k
>0,
no
larg
er t
han
a g
iven
max
imu
m s
ize,
p, s
o a
s to
min
imiz
e th
e to
tal c
ost
of
the
edg
es c
ut.
�D
efin
e :
as a
wei
gh
ted
co
nn
ecti
vity
mat
rix
des
crib
ing
th
e ed
ges
of
G.
�A
k-w
ay p
arti
tio
no
f G
is a
set
of
no
n-e
mp
ty, p
airw
ise-
dis
join
t
sub
sets
of
G, v
1,…
,vk,
su
ch t
hat
�A
par
titi
on
is s
aid
to
be
adm
issi
ble
if
�P
rob
lem
:F
ind
a m
inim
al-c
ost
per
mis
sib
le p
arti
tio
n o
f G
01
<≤
=w
pi
ni
,,
,L
Cc
ij
nij
==
(),
,,
,1L
vG
iik =
=1
U
||
,,
,v
pi
ki
≤=
1L
EE
244
29
Ho
w b
ig is
th
e se
arch
sp
ace?
�n
no
des
, ksu
bse
ts o
f si
ze p
such
th
at k
p=
n
�w
ays
to c
ho
ose
th
e fi
rst
sub
set
�w
ays
to c
ho
ose
th
e se
con
d, e
tc.
�w
ays
tota
l
�n
=40
, p=
10
�In
gen
eral
, so
lvin
g p
rob
lem
s w
her
e
are
imp
ract
ical
fo
r re
al c
ircu
its
(>1,
000,
000
gat
es)
()n p
np
p−
12
k
n pn
p
p
p p
p p!
−
L
>1
02
0
Tn
n∝
>β
β,2
EE
244
30
Heu
rist
ics
for
n-W
ay P
arti
tio
nin
g�
Har
d p
rob
lem
an
d n
o r
eally
go
od
heu
rist
ics
for
n>2
�D
irec
t M
eth
od
s:S
tart
wit
h s
eed
no
de
for
each
par
titi
on
an
d
assi
gn
no
des
to
eac
h p
arti
tio
n u
sin
g s
om
e cr
iter
ion
(e.
g. s
um
of
wei
gh
ted
co
nn
ecti
on
s in
to p
arti
tio
n)
�G
rou
p M
igra
tio
n M
eth
od
s:S
tart
wit
h (
ran
do
m)
init
ial p
arti
tio
n
and
mig
rate
no
des
am
on
g p
arti
tio
ns
via
som
e h
euri
stic
�M
etri
c A
lloca
tio
n M
eth
od
s: u
ses
met
rics
oth
er t
han
co
nn
ecti
on
g
rap
h a
nd
th
en c
lust
ers
no
des
bas
ed o
n m
etri
c o
ther
th
an
exp
licit
co
nn
ecti
vity
.
�S
toch
asti
c O
pti
miz
atio
n A
pp
roac
hes
:U
se a
gen
eral
-pu
rpo
se
sto
chas
tic
app
roac
h li
ke s
imu
late
d a
nn
ealin
g o
r g
enet
ic
alg
ori
thm
s
�U
sual
ly a
pp
ly t
wo
-way
par
titi
on
ing
(K
ern
igh
an-L
in o
r F
idu
ccia
-M
ath
eyse
s) r
ecu
rsiv
ely,
or
in s
om
e ca
ses
sim
ula
ted
an
nea
ling
EE
244
31
Par
titi
on
ing
: R
and
om
plu
s Im
pro
vem
ent
�R
and
om
Par
titi
on
s, S
ave
Bes
t to
Dat
e
�F
ast,
bu
t ca
n b
e sh
ow
n t
o b
e O
(n2 )
�F
ew o
pti
mal
or
nea
r o
pti
mal
so
luti
on
s, h
ence
low
pro
bab
ility
of
fin
din
g o
ne
e.g
. 2-w
ay p
arti
tio
n o
f 0-
1 w
eig
ht
gra
ph
s w
ith
32
no
des
, ~3-
5
op
tim
al p
arti
tio
ns
ou
t o
f(
)1 2
32
16
10
7
)
on
an
y t
rial
⇒<
−P
success
(
EE
244
32
Par
titi
on
ing
: M
ax-f
low
, Min
-cu
t
�M
ax-f
low
, Min
-cu
t: u
sefu
l fo
r u
nco
nst
rain
ed lo
wer
bo
un
d
�F
ord
& F
ulk
erso
n, “
Flo
ws
in N
etw
ork
s,”
Pri
nce
ton
Un
iv. P
ress
, 196
2
�E
dg
e w
eig
hts
of
G c
orr
esp
on
d t
o m
axim
um
flo
w c
apac
itie
s b
etw
een
pai
rs o
f n
od
es
�C
ut
is a
sep
arat
ion
of
no
des
into
tw
o d
isjo
int
sub
sets
; cu
t ca
pac
ity
is
the
cost
of
a p
arti
tio
n
Max
-flo
w M
in-c
ut
Th
eore
m:T
he
max
imu
m f
low
bet
wee
n a
ny
pai
r o
f n
od
es =
the
min
imu
m c
ut
cap
acit
y o
f al
l cu
ts w
hic
h s
epar
ate
the
two
no
des
Co
mp
uti
ng
max
-flo
w t
hro
ug
h g
rap
h is
pro
bab
ly t
oo
exp
ensi
ve
EE
244
33
Tw
o-W
ay P
arti
tio
nin
g
(Ker
nig
han
& L
in)
�C
on
sid
er t
he
set
So
f 2n
vert
ices
, all
of
equ
al s
ize
for
no
w,
wit
h a
n a
sso
ciat
ed c
ost
mat
rix
�A
ssu
me
Cis
sym
met
ric
and
�W
e w
ant
to p
arti
tio
n S
into
tw
o s
ub
sets
Aan
d B
, eac
h w
ith
np
oin
ts, s
uch
th
at t
he
exte
rnal
co
st
is m
inim
ized
�S
tart
wit
h a
ny
arb
itra
ry p
arti
tio
n [
A,B
] o
f S
and
try
to
d
ecre
ase
the
init
ial c
ost
Tb
y a
seri
es o
f in
terc
han
ges
of
sub
sets
of
Aan
d B
�W
hen
no
fu
rth
er im
pro
vem
ent
is p
oss
ible
, th
e re
sult
ing
p
arti
tio
n [
A’,B
’] is
alo
cal m
inim
um
(an
d h
as s
om
e p
rob
abili
ty o
f b
ein
g a
glo
bal
min
imu
m w
ith
th
is s
chem
e)
�(B
e su
re t
o t
ake
a m
om
ent
to t
alk
abo
ut
loca
l an
d g
lob
al
min
ima)
Cc
ij
nij
==
(),
,,
,1
2L
ci
ii=
∀0
TC
ab
AB
=∑
×
EE
244
34
Ker
nig
han
& L
in:
Val
ue
of
a co
nfi
gu
rati
on
�F
or
each
vert
ex a
in p
arti
tio
n A
:
�ex
tern
al c
ost
(co
mp
ute
d t
he
sam
e fo
r E
b)
�in
tern
al c
ost
(c
om
pu
ted
th
e sa
me
for
Ib)
�F
or
each
ver
tex
z in
th
e se
t S
, th
e d
iffe
ren
ce (
D)
bet
wee
n e
xter
nal
(E
) an
d in
tern
al (
I) c
ost
s is
giv
en b
y:
aA
∈
Ec
aa
yy
B
=∑ ∈
Ic
aa
xx
A
=∑ ∈
DE
Iz
Sz
zz
=−
∀∈
EE
244
35
Ker
nig
han
& L
in:
Val
ue
of
on
e sw
ap
�F
or
each
:
�ex
tern
al c
ost
(sam
e fo
r E
b)
�in
tern
al c
ost
(s
ame
for
Ib)
�If
a ∈ ∈∈∈
Α
Α
Α
Α a
nd
b ∈ ∈∈∈
Β
Β
Β
Β a
re in
terc
han
ged
, th
en t
he
gai
n:
�P
roo
f: If
Zis
th
e to
tal c
ost
of
con
nec
tio
ns
bet
wee
n p
arti
tio
ns
Aan
d B
, exc
lud
ing
ver
tice
s a
and
b, t
hen
:
aA
∈
Ec
aa
yy
B
=∑ ∈
Ic
aa
xx
A
=∑ ∈
DE
Iz
Sz
zz
=−
∀∈
gD
Dc
ab
ab
=+
−2
TZ
EE
c
TZ
II
cg
ain
TT
DD
ca
ba
ba
b
ba
ab
ab
ab
ba
ab
ab
, ,,
,
=+
+−
=+
++
=
−=
+−
2
EE
244
36
Ker
nig
han
& L
in:
Ch
oo
sin
g s
wap
(1)
Co
mp
ute
all
Dva
lues
in S
(2)
Ch
oo
se a
i, b
isu
ch t
hat
is m
axim
ized
(3)
Set
aian
db
ias
ide
and
cal
l th
em a
i’an
d b
i’
(4)
Rec
alcu
late
th
e D
val
ues
fo
r al
l th
e el
emen
ts o
f
AB
a
b
ji
ji
ba
ba
ic
DD
g2
−+
=
Aa
Bb
ij
−−
{}
,{
}
DD
cc
xA
a
DD
cc
yB
b
xx
xa
xbi
yy
yb
yaj
ij
ji
' '
,{
}
,{
}
=+
−∈
−
=+
−∈
−
22
22
EE
244
37
Ker
nig
han
& L
in:
Par
titi
on
ing
Alg
ori
thm
Alg
ori
thm
KL
(G, g
rap
h o
f 2N
no
des
)
Init
ializ
e -
crea
te in
itia
l bi-
par
titi
on
into
A, B
each
of
N n
od
es
/* C
om
pu
te g
lob
al v
alu
e o
f in
div
idu
al s
wap
s o
f n
od
es *
/
Rep
eat
un
til n
o f
urt
her
imp
rove
men
t{
for
I = 1
to
N d
o{
fin
d p
air
of
un
lock
ed n
od
es a
iin
A a
nd
bi i
n B
wh
ose
exc
han
ge
lead
s to
larg
est
dec
reas
e o
r sm
alle
st in
crea
se in
co
st
cost
_i=
ch
ang
e in
co
st d
ue
to e
xch
ang
ing
ai
and
bi
lock
do
wn
ai
and
bi
so t
hey
do
n’t
par
tici
pat
e in
fu
ture
mo
ves
}
/* f
ind
wh
ich
seq
uen
ce o
f sw
aps
gav
e th
e b
est
resu
lt *
/
fin
d l
such
th
at s
um
of
cost
(1<=
l) is
max
imiz
ed
mo
ve a
i0<
=l f
rom
Ato
B
mo
ve b
i 0<
=l f
rom
Bto
A
}
EE
244
38
Tw
o-W
ay P
arti
tio
nin
g
(Ker
nig
han
& L
in)
�F
ind
po
int
(ex
chan
ge)
mat
wh
ich
cu
mu
lati
veg
ain
max
imiz
ed
�P
erfo
rm e
xch
ang
es 1
th
rou
gh
m
�W
hat
is t
he
tim
e an
d m
emo
ry c
om
ple
xity
of
this
alg
ori
thm
?
gk
k
i =∑1
i1
23
mn
Cum
ulat
ive
gain
Cum
ulat
ive
gain
EE
244
39
Ker
nig
han
-Lin
(K
L)
Exa
mp
le -
1K
ern
igh
an-L
in (
KL
) E
xam
ple
-1
a b c d
e f g h
0--
05
Ste
p N
o.
Vert
ex Pair
Gain
Cut-
cost
[©S
arr
afz
ad
eh
]
EE
244
40
Ker
nig
han
-Lin
(K
L)
Exa
mp
le -
2K
ern
igh
an-L
in (
KL
) E
xam
ple
-2
a b cd ddde
f
g gggh
0--
05
1{ d, g }
32
Ste
p N
o.
Vert
ex Pair
Gain
Cut-
cost
[©S
arr
afz
ad
eh
]
EE
244
41
Ker
nig
han
-Lin
(K
L)
Exa
mp
le -
3K
ern
igh
an-L
in (
KL
) E
xam
ple
-3
a
bc ccc
d ddde
f fff
g gggh
0--
05
1{ d, g }
32
2{ c, f
}1
1
Ste
p N
o.
Vert
ex Pair
Gain
Cut-
cost
[©S
arr
afz
ad
eh
]
EE
244
42
Ker
nig
han
-Lin
(K
L)
Exa
mp
le -
fin
ish
Ker
nig
han
-Lin
(K
L)
Exa
mp
le -
fin
ish
4{ a, e }
-2
5
0--
05
1{ d, g }
32
2{ c, f
}1
1
3{ b, h }
-2
3
Ste
p N
o.
Vert
ex Pair
Gain
Cut-
cost
[©S
arr
afz
ad
eh
]
a
bc ccc
d ddde
f fff
g gggh
EE
244
43
Tim
e C
om
ple
xity
of
K-L
Par
titi
on
ing
Tim
e C
om
ple
xity
of
K-L
Par
titi
on
ing
�A
pas
s is
a s
et o
f o
per
atio
ns
nee
ded
to
fin
d e
xch
ang
e se
ts
�In
itia
l dif
fere
nce
vec
tor
D c
om
pu
tati
on
is n
2
�U
pd
ate
of
D a
fter
lock
ing
a p
air
(w
e lo
ck d
ow
n o
ne
mo
re
each
pas
s)
�(n
-1)+
(n-2
)+…
+2+
1 � ���
n2
�D
om
inan
t ti
me
fact
or
–se
lect
ion
of
the
nex
t p
air
to
exch
ang
e
�N
eed
to
so
rt D
val
ues
�S
ort
ing
is n
*lo
g(n
)
�(n
)lo
g(n
)+(n
-1)l
og
(n-1
)+(n
-2)+
…+
2lo
g2 � ���
n2 l
og
n
�T
ota
l tim
e is
n2 lo
g n
�A
pas
s is
a s
et o
f o
per
atio
ns
nee
ded
to
fin
d e
xch
ang
e se
ts
�In
itia
l dif
fere
nce
vec
tor
D c
om
pu
tati
on
is n
2
�U
pd
ate
of
D a
fter
lock
ing
a p
air
(w
e lo
ck d
ow
n o
ne
mo
re
each
pas
s)
�(n
-1)+
(n-2
)+…
+2+
1 � ���
n2
�D
om
inan
t ti
me
fact
or
–se
lect
ion
of
the
nex
t p
air
to
exch
ang
e
�N
eed
to
so
rt D
val
ues
�S
ort
ing
is n
*lo
g(n
)
�(n
)lo
g(n
)+(n
-1)l
og
(n-1
)+(n
-2)+
…+
2lo
g2 � ���
n2 l
og
n
�T
ota
l tim
e is
n2 lo
g n
EE
244
44
Just
wh
at d
oes
par
titi
on
ing
do
?Ju
st w
hat
do
es p
arti
tio
nin
g d
o?
�R
edu
ces
the
pro
ble
m s
ize
enab
ling
a “
div
ide
and
con
qu
er”
app
roac
h t
o p
rob
lem
so
lvin
g
�N
atu
rally
evo
lves
th
e n
etlis
t to
war
d a
fu
ll p
lace
men
t
�R
edu
ces
the
pro
ble
m s
ize
enab
ling
a “
div
ide
and
con
qu
er”
app
roac
h t
o p
rob
lem
so
lvin
g
�N
atu
rally
evo
lves
th
e n
etlis
t to
war
d a
fu
ll p
lace
men
t
Wh
ere
do
es p
arti
tio
nin
g f
it in
?W
her
e d
oes
par
titi
on
ing
fit
in?
EE
244
46
Par
titi
on
ing
Par
titi
on
ing
�In
GO
RD
IAN
, par
titi
on
ing
is u
sed
to
co
nst
rain
t th
e
mo
vem
ent
of
mo
du
les
rath
er t
han
red
uce
pro
ble
m s
ize
�B
y p
erfo
rmin
g p
arti
tio
nin
g, w
e ca
n it
erat
ivel
y im
po
se a
new
set
of
con
stra
ints
on
th
e g
lob
al o
pti
miz
atio
n p
rob
lem
�A
ssig
n m
od
ule
s to
a p
arti
cula
r b
lock
�P
arti
tio
nin
g is
det
erm
ined
by
�R
esu
lts
of
glo
bal
pla
cem
ent
�S
patia
l (x,
y) d
istr
ibut
ion
of m
odul
es
�P
arti
tio
nin
g c
ost
�W
ant a
min
-cut
par
titio
n
�In
GO
RD
IAN
, par
titi
on
ing
is u
sed
to
co
nst
rain
t th
e
mo
vem
ent
of
mo
du
les
rath
er t
han
red
uce
pro
ble
m s
ize
�B
y p
erfo
rmin
g p
arti
tio
nin
g, w
e ca
n it
erat
ivel
y im
po
se a
new
set
of
con
stra
ints
on
th
e g
lob
al o
pti
miz
atio
n p
rob
lem
�A
ssig
n m
od
ule
s to
a p
arti
cula
r b
lock
�P
arti
tio
nin
g is
det
erm
ined
by
�R
esu
lts
of
glo
bal
pla
cem
ent
�S
patia
l (x,
y) d
istr
ibut
ion
of m
odul
es
�P
arti
tio
nin
g c
ost
�W
ant a
min
-cut
par
titio
n
EE
244
47
Par
titi
on
ing
du
e to
Glo
bal
Op
tim
izat
ion
Par
titi
on
ing
du
e to
Glo
bal
Op
tim
izat
ion
�S
ort
th
e m
od
ule
s b
y th
eir
x co
ord
inat
e (f
or
a ve
rtic
al
cut)
�C
ho
ose
a c
ut
line
such
th
at
�S
ort
th
e m
od
ule
s b
y th
eir
x co
ord
inat
e (f
or
a ve
rtic
al
cut)
�C
ho
ose
a c
ut
line
such
th
at
→ →→→p
pp
MM
M,
'''
∈ ∈∈∈
∑ ∑∑∑∑ ∑∑∑
∈ ∈∈∈∈ ∈∈∈
≈ ≈≈≈= ===
∈ ∈∈∈≤ ≤≤≤
Mu
uM
uu
pp
uu
FF
Mu
Mu
xx
pp
α ααα0.5
'',
'
''''
''
''
∈ ∈∈∈
Par
titi
on
ing
Imp
rove
men
t -
IP
arti
tio
nin
g Im
pro
vem
ent
-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
Lay
ou
t af
ter
Min
-cu
tL
ayo
ut
afte
r M
in-c
ut
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
EE
244
50
Th
ou
gh
ts o
n P
arti
tio
nin
gT
ho
ug
hts
on
Par
titi
on
ing
Sti
ll an
act
ive
area
of
rese
arch
�R
esu
lts
hig
hly
dep
end
ent
on
heu
rist
ic
imp
rove
men
ts a
nd
co
nte
xt
Par
titi
on
ing
is t
he
wo
rkh
ors
e o
f p
lace
men
t an
d
flo
orp
lan
nin
g
�A
s a
resu
lt p
arti
tio
nin
gs
mu
st b
e ve
ry f
ast
�A
lot
of
was
ted
aca
dem
ic e
ffo
rt o
n s
low
(b
ut
slig
htl
y b
ette
r) p
arti
tio
nin
g a
pp
roac
hes
K&
L, F
&M
hav
e ea
ch h
eld
up
ver
y w
ell
Sti
ll an
act
ive
area
of
rese
arch
�R
esu
lts
hig
hly
dep
end
ent
on
heu
rist
ic
imp
rove
men
ts a
nd
co
nte
xt
Par
titi
on
ing
is t
he
wo
rkh
ors
e o
f p
lace
men
t an
d
flo
orp
lan
nin
g
�A
s a
resu
lt p
arti
tio
nin
gs
mu
st b
e ve
ry f
ast
�A
lot
of
was
ted
aca
dem
ic e
ffo
rt o
n s
low
(b
ut
slig
htl
y b
ette
r) p
arti
tio
nin
g a
pp
roac
hes
K&
L, F
&M
hav
e ea
ch h
eld
up
ver
y w
ell
EE
244
51
Rev
iew
ing
ou
r G
ener
al P
roce
du
reR
evie
win
g o
ur
Gen
eral
Pro
ced
ure
�T
ake
a re
al w
orl
d p
rob
lem
–p
arti
tio
nin
g o
f n
etlis
ts
�C
ast
in a
mat
hem
atic
al a
bst
ract
ion
–th
is o
ften
req
uir
es
sim
plif
icat
ion
�Id
enti
fy c
ost
fu
nct
ion
to
be
op
tim
ized
�Id
enti
fy s
ize
of
sear
ch s
pac
e
�Is
glo
bal
op
tim
alit
y co
mp
uta
tio
nal
ly f
easi
ble
?
�Y
es –
go
to
it!
�N
o –
�Id
enti
fy h
euri
stic
s th
at a
pp
roxi
mat
e g
lob
al o
pti
mu
m
�S
imp
lify
pro
ble
m f
urt
her
an
d s
ee if
yo
u c
an a
chie
ve a
loca
l op
tim
um
in a
co
mp
uta
tio
nal
ly e
ffic
ien
t m
ann
er
�P
lug
bac
k in
th
e o
rig
inal
pro
ble
m a
nd
see
ho
w it
wo
rks
�T
ake
a re
al w
orl
d p
rob
lem
–p
arti
tio
nin
g o
f n
etlis
ts
�C
ast
in a
mat
hem
atic
al a
bst
ract
ion
–th
is o
ften
req
uir
es
sim
plif
icat
ion
�Id
enti
fy c
ost
fu
nct
ion
to
be
op
tim
ized
�Id
enti
fy s
ize
of
sear
ch s
pac
e
�Is
glo
bal
op
tim
alit
y co
mp
uta
tio
nal
ly f
easi
ble
?
�Y
es –
go
to
it!
�N
o –
�Id
enti
fy h
euri
stic
s th
at a
pp
roxi
mat
e g
lob
al o
pti
mu
m
�S
imp
lify
pro
ble
m f
urt
her
an
d s
ee if
yo
u c
an a
chie
ve a
loca
l op
tim
um
in a
co
mp
uta
tio
nal
ly e
ffic
ien
t m
ann
er
�P
lug
bac
k in
th
e o
rig
inal
pro
ble
m a
nd
see
ho
w it
wo
rks
EE
244
52
Bac
k in
th
e R
TL
Des
ign
Flo
w
RT
LS
yn
thesis
HD
L
netl
ist
log
ico
pti
miz
ati
on
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
Mo
du
leG
en
era
tors
Man
ual
Desig
n
EE
244
53
Fo
r N
ext
Cla
ssF
or
Nex
t C
lass
�R
ead
th
e F
idu
ccia
& M
atth
eyse
sp
aper
�R
ead
th
e G
ord
ian
pap
er
�R
ead
th
e F
idu
ccia
& M
atth
eyse
sp
aper
�R
ead
th
e G
ord
ian
pap
er
EE
244
54
Ext
ra S
lides
Ext
ra S
lides
�S
imu
late
d a
nn
ealin
g
�F
idu
ccia
& M
atth
eyse
s
�S
imu
late
d a
nn
ealin
g
�F
idu
ccia
& M
atth
eyse
s
EE
244
55
Sim
ula
ted
An
nea
ling
Sim
ula
ted
An
nea
ling
�U
ses a
nalo
gy w
ith
meta
llu
rgic
al
an
nealin
g
�S
tart
wit
h a
ran
do
m in
itia
l p
art
itio
nin
g
�G
en
era
te a
new
part
itio
nin
g b
y e
xch
an
gin
g t
wo
ra
nd
om
ly c
ho
sen
co
mp
on
en
ts f
rom
part
1 a
nd
p
art
2
�C
om
pu
te t
he c
han
ge
in
sco
re:
�If
,
a lo
wer
en
erg
y s
tate
is f
ou
nd
, th
e m
ove i
s
acc
ep
ted
�If
, th
e m
ove i
s a
ccep
ted
wit
h p
rob
ab
ilit
y
, w
here
t is “
tem
pera
ture
”
�T
em
pera
ture
, t,
is s
low
ly r
ed
uc
ed
�H
elp
s a
vo
id lo
cal m
inim
a
�U
ses a
nalo
gy w
ith
meta
llu
rgic
al
an
nealin
g
�S
tart
wit
h a
ran
do
m in
itia
l p
art
itio
nin
g
�G
en
era
te a
new
part
itio
nin
g b
y e
xch
an
gin
g t
wo
ra
nd
om
ly c
ho
sen
co
mp
on
en
ts f
rom
part
1 a
nd
p
art
2
�C
om
pu
te t
he c
han
ge
in
sco
re:
�If
,
a lo
wer
en
erg
y s
tate
is f
ou
nd
, th
e m
ove i
s
acc
ep
ted
�If
, th
e m
ove i
s a
ccep
ted
wit
h p
rob
ab
ilit
y
, w
here
t is “
tem
pera
ture
”
�T
em
pera
ture
, t,
is s
low
ly r
ed
uc
ed
�H
elp
s a
vo
id lo
cal m
inim
a
s0
δ<
s0
δ≥
exp
(s
/t)
−δ
sδ
EE
244
56
Tw
o-W
ay P
arti
tio
nin
g
(Fid
ucc
ia&
Mat
they
ses)
�M
ove
on
e ce
ll at
a t
ime
fro
m o
ne
sid
e o
f th
e
par
titi
on
to
th
e o
ther
in a
n a
ttem
pt
to m
inim
ize
the
cuts
eto
f th
e fi
nal
par
titi
on
�b
ase
cell
--ce
ll to
be
mo
ved
�g
ain
g(i
)--
no
. of
net
s b
y w
hic
h t
he
cuts
etw
ou
ld
dec
reas
e if
cel
l i w
ere
mo
ved
fro
m p
arti
tio
n A
to p
arti
tio
n
B(m
ay b
e n
egat
ive)
�T
o p
reve
nt
thra
shin
g, o
nce
a c
ell i
s m
ove
d it
is
lock
ed f
or
an e
nti
re p
ass
�C
laim
is O
(n)
tim
e
EE
244
57
Tw
o-W
ay P
arti
tio
nin
g
(Fid
ucc
ia&
Mat
they
ses)
�S
tep
s:
(1)
Ch
oo
se a
cel
l
(2)
Mo
ve it
(3)
Up
dat
e th
e g
(i)’
s o
f th
e n
eig
hb
ors
EE
244
58
Tw
o-W
ay P
arti
tio
nin
g
(Fid
ucc
ia&
Mat
they
ses)
�If
p(i
)=
no
. of
pin
s o
n c
ell i
:
�B
in-s
ort
cel
ls o
n g
i
�T
ime
req
uir
ed t
o m
ain
tain
eac
h b
uck
et a
rray
O(P
)/p
ass
−<
<p
ig
pi
i(
)(
)
-pm
ax
pm
ax
MA
X_G
AIN
LO
CK
ED
_CE
LL
S
......
CE
LL
1 2
3C
EE
244
59
Tw
o-W
ay P
arti
tio
nin
g
(Fid
ucc
ia&
Mat
they
ses)
�M
ove
th
e C
ell
(1)
Fin
d t
he
firs
t ce
ll o
f h
igh
est
gai
n t
hat
is n
ot
lock
ed a
nd
su
ch t
hat
mo
vin
g it
wo
uld
no
t ca
use
an
imb
alan
ce
�B
reak
tie
by
cho
osi
ng
th
e o
ne
that
giv
es t
he
bes
t b
alan
ce
(2)
Ch
oo
se t
his
as
the
bas
e ce
ll. R
emo
ve it
fro
m t
he
bu
cket
list
and
pla
ce it
on
th
e L
OC
KE
D li
st. U
pd
ate
it t
o t
he
oth
er p
arti
tio
n.
�U
pd
atin
g C
ell G
ain
s
Cri
tica
l net
�G
iven
a p
arti
tio
n (
A|B
), w
e d
efin
e th
e d
istr
ibu
tio
n o
f n
as a
n
ord
ered
pai
r o
f in
teg
ers
(A(n
),B
(n))
, wh
ich
rep
rese
nts
th
e
nu
mb
er o
f ce
lls n
et n
has
in b
lock
s A
and
Bre
spec
tive
ly (
can
be
com
pu
ted
in O
(P)
tim
e fo
r al
l net
s)
EE
244
60
Tw
o-W
ay P
arti
tio
nin
g
(Fid
ucc
ia&
Mat
they
ses)
�N
et is
cri
tica
lif
ther
e ex
ists
a c
ell o
n it
su
ch t
hat
if it
w
ere
mo
ved
it w
ou
ld c
han
ge
the
net
’s c
ut
stat
e(w
het
her
it is
cu
t o
r n
ot)
.
�N
et is
cri
tica
l if
A(n
)=0,
1o
r B
(n)=
0,1
�G
ain
of
cell
dep
end
s o
nly
on
its
crit
ical
net
s:�
If a
net
is n
ot
crit
ical
, its
cu
tsta
teca
nn
ot
be
affe
cted
by
the
mo
ve
�A
net
wh
ich
is n
ot
crit
ical
eit
her
bef
ore
or
afte
r a
mo
ve
can
no
t in
flu
ence
th
e g
ain
s o
f it
s ce
lls
�T
his
is t
he
bas
is o
f th
e lin
ear-
tim
e cl
aim
EE
244
61
Tw
o-W
ay P
arti
tio
nin
g
(Fid
ucc
ia&
Mat
they
ses)
�L
et F
be
the
fro
mp
arti
tio
n o
f ce
ll ia
nd
Tth
e to
par
titi
on
�g
(i)
= F
S(i
) -
TE
(i),
wh
ere:
�F
S(i
) =
no
. of
net
s w
hic
h h
ave
cell
ias
thei
r o
nly
Fce
ll
�T
E(i
)=
no
. of
net
s w
hic
h c
on
tain
ian
d h
ave
an e
mp
ty T
sid
e
Fi
ba
T
FS
(i)
TE
(i)
EE
244
62
Tw
o-W
ay P
arti
tio
nin
g
(Fid
ucc
ia&
Mat
they
ses)
�C
om
pu
te t
he
init
ial g
ain
s o
f al
l un
lock
ed c
ells
:fo
reach
(fre
ecell i)
{
g(i
) =
0;
F =
th
e “
fro
m”
part
itio
n o
f cell i;
T =
th
e “
to”
part
itio
n o
f cell i;
fore
ach
(net
n o
n c
ell i)
{
if(F
(n)
= 1
) g
(i)+
+;
if(T
(n)
= 0
) g
(i)-
-;
}
}
�R
equ
ires
O(P
) w
ork
to
inti
aliz
e
�n
et is
cri
tica
l bef
ore
th
e m
ove
iff
F(n
)=1
or
T(n
)=0
or
T(n
) =
1
�F
(n)
=0
do
es n
ot
occ
ur
bec
ause
bas
e ce
ll o
n F
sid
e b
efo
re
�n
et is
cri
tica
l aft
er t
he
mo
ve if
fT
(n)=
1 o
r F
(n)=
0 o
r F
(n)=
1
�T
(n)
=0
do
es n
ot
occ
ur
bec
ause
bas
e ce
ll o
n T
sid
e af
ter
EE
244
63
Tw
o-W
ay P
arti
tio
nin
g
(Fid
ucc
ia&
Mat
they
ses)
�M
ain
loo
p:
lock b
ase c
ell;
fore
ach
(net
n o
n b
ase c
ell)
{
if(T
(n)
==
0)
incre
men
t g
ain
s o
f all f
ree c
ells o
n n
et
n;
els
e if(
T(n
) =
= 1
) d
ecre
men
t g
ain
s o
f th
e T
cell o
n n
et
n
if it
is f
ree;
F(n
)--;
T(n
)++
;
/* c
heck c
riti
cal n
ets
aft
er
the m
ove *
/
if(F
(n)=
= 0
) d
ecre
men
t g
ain
s o
f all f
ree c
ells o
n n
et
n;
els
e if(
F(n
) =
= 1
) in
cre
men
t g
ain
of
the o
nly
F c
ell o
n
net
n if
it is f
ree;
}
�T
ime
com
ple
xity
O(n
log
(n))
?
EE
244
64
Ker
nig
han
-Lin
(K
L)
Exa
mp
le -
fin
ish
Ker
nig
han
-Lin
(K
L)
Exa
mp
le -
fin
ish
a b c d
e f g h
4{ a, e }
-2
5
0--
05
1{ d, g }
32
2{ c, f
}1
1
3{ b, h }
-2
3
Ste
p N
o.
Vert
ex Pair
Gain
Cut-
cost
[©S
arr
afz
ad
eh
]
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