Partitioning for Physical Design Prof. A. R. Newton …Partitioning for Physical Design Prof. A. R....
Transcript of Partitioning for Physical Design Prof. A. R. Newton …Partitioning for Physical Design Prof. A. R....
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
]