routing in integrated circuits a. kahng k. keutzer a. r ... · routing in integrated circuits a....
TRANSCRIPT
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
Kah
ng
/Keu
tzer/
New
ton
Ro
uti
ng
in
In
teg
rate
d C
ircu
its
A. K
ah
ng
K. K
eu
tzer
A. R
. N
ew
ton
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 2
Kah
ng
/Keu
tzer/
New
ton
Cla
ss
Ne
ws
�W
ednesday:
�P
resenta
tion o
f re
searc
h topic
s in p
lacem
ent, r
outing, and tim
ing
�E
ach g
roup w
ill d
o a
short
pre
senta
tion a
t th
e w
hiteboard
:
-In
troduce g
roup m
em
bers
-D
escribe y
our
researc
h p
roble
m
-N
am
e m
ento
rs
-N
am
e r
esourc
es y
ou w
ill n
eed to d
o p
roje
ct and w
here
you
will
get th
em
-D
escribe y
our
appro
ach –
meth
odolo
gic
ally
and c
onte
nt
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 3
Kah
ng
/Keu
tzer/
New
ton
To
da
y -
Ima
gin
e
�Y
ou h
ave to p
lan t
ransport
ation f
or
a n
ew
city the s
ize o
f C
hic
ago
�M
any d
we
llin
gs n
ee
d d
irect
road
s that
can’t b
e u
sed b
y
anyo
ne e
lse
�Y
ou c
an a
ffect th
e layo
ut
of house
s a
nd n
eig
hborh
oo
ds
but
the a
rchitects
and p
lan
ners
will
com
pla
in
�A
nd …
you
’re t
old
that th
e t
ime a
long a
ny p
ath
can’t b
e
long
er
than a
fix
ed a
mount
�W
hat
are
som
e o
f your
consid
era
tions?
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 4
Kah
ng
/Keu
tzer/
New
ton
Wh
at
are
so
me
of
yo
ur
co
ns
ide
rati
on
s?
�H
ow
many leve
ls d
o m
y r
oads n
eed t
o g
o? R
em
em
ber:
H
igh
er
is m
ore
expe
nsiv
e.
�H
ow
do I
avoid
cong
estion?
�W
hat
basic
str
uctu
re d
o I w
ant fo
r m
y r
oads?
�M
anhattan?
�C
hic
ago?
�B
osto
n?
�A
uto
mate
d r
oute
tools
have t
o s
olv
e p
roble
ms o
f com
para
ble
com
ple
xity o
n e
very
lead
ing e
dg
e c
hip
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 5
Kah
ng
/Keu
tzer/
New
ton
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
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
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 6
Kah
ng
/Keu
tzer/
New
ton
Ph
ysic
al D
esig
n F
low
Rea
d N
etlis
t
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
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 7
Kah
ng
/Keu
tzer/
New
ton
Ro
uti
ng
Ap
pli
ca
tio
ns
Blo
ck-b
ased
Blo
ck-b
ased
Mix
ed
Cell a
nd
Blo
ck
Mix
ed
Cell a
nd
Blo
ck
Cell-b
ased
Cell-b
ased
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 8
Kah
ng
/Keu
tzer/
New
ton
Ce
ll base
d p
lacem
ent le
aves u
s w
ith …
(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
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 9
Kah
ng
/Keu
tzer/
New
ton
Ro
uti
ng
Alg
ori
thm
s
Hard
to tackle
hig
h-level is
sues lik
e c
ongestion
and w
ire-p
lannin
g a
nd low
level deta
ils o
f pin
-connection a
t th
e s
am
e tim
e
�G
lobal ro
uting
�Id
entify
routing r
esourc
es t
o b
e u
sed
�Id
entify
layers
(and tra
cks)
to b
e u
sed
�A
ssig
n p
art
icu
lar
nets
to t
hese r
esourc
es
�A
lso u
sed in f
loorp
lan
nin
g a
nd p
lacem
ent
�D
eta
il ro
uting
�A
ctu
ally
de
fine p
in-t
o-p
in c
onn
ections
�M
ust
unders
tand m
ost or
all
desig
n r
ule
s
�M
ay u
se a
com
pacto
r to
optim
ize r
esult
�N
ecessary
in a
ll a
pp
lications
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
0
Kah
ng
/Keu
tzer/
New
ton
Ba
sic
Ru
les
of
Ro
uti
ng
-1
Ph
oto
co
urt
es
y:
Ja
n M
. R
ab
ae
yA
na
nth
aC
ha
nd
rakasa
nB
ori
vo
je N
iko
lic
�W
irin
g/r
outing
perf
orm
ed in layers
–5-9
(-1
1),
typic
ally
only
in “
Manhattan”
N/S
E/W
directions
�E
.g.
layer
1 –
N/S
�La
yer
2 –
E/W
�A
segm
ent cannot
cro
ss a
noth
er
segm
ent on the s
am
e
wirin
g layer
or
…?
�W
ire s
egm
ents
can
cro
ss w
ires o
n o
ther
layers
�P
ow
er
and g
round
may h
ave their o
wn
layers
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
1
Kah
ng
/Keu
tzer/
New
ton
Ba
sic
Ru
les
of
Ro
uti
ng
–P
art
2
�R
outin
g c
an b
e o
n a
fix
ed g
rid –
�C
ase 1
: D
eta
iled r
outing o
nly
in c
han
ne
ls
�W
irin
g c
an o
nly
go o
ver
a r
ow
of cells
when there
is a
fr
ee tra
ck –
can b
e insert
ed w
ith a
“fe
edth
rough”
�D
esig
n m
ay u
se o
f m
eta
l-1, m
eta
l-2
�C
ells
mustbring s
ignals
(i.e. in
puts
, outp
uts
) out to
the
channel th
rough “
port
s”
or
“pin
s”
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
2
Kah
ng
/Keu
tzer/
New
ton
Ba
sic
Ru
les
of
Ro
uti
ng
–P
art
3
�R
outin
g c
an b
e o
n a
fix
ed o
r grid
less
(aka
are
a
routing)
�C
ase 1
: D
eta
iled r
outing o
ver
cells
�W
irin
g c
an g
o o
ver
cells
�D
esig
n o
f cells
must tr
y to m
inim
ize o
bsta
cle
s to
routing –
I.e. m
inim
ize u
se o
f m
eta
l-1, m
eta
l-2
�C
ells
do n
otneed to b
ring s
ignals
(i.e. in
puts
, outp
uts
) out to
the c
hannel –
the r
oute
will
com
e to them
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
3
Kah
ng
/Keu
tzer/
New
ton
Ta
xo
no
my o
f V
LS
I R
ou
ters
Gra
ph
Searc
h
Ste
iner
Itera
tiv
e
Hie
rarc
hic
al
Gre
ed
yL
eft
-Ed
ge
Riv
er
Sw
itch
bo
x
Ch
an
nel
Maze
Lin
e P
rob
e
Lin
e E
xp
ansi
on
Res
tric
ted
Gen
era
l P
urp
ose
Po
wer
& G
rou
nd
Clo
ck
Glo
bal
Deta
iled
Sp
ecia
lize
d
Ro
ute
rs
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
4
Kah
ng
/Keu
tzer/
New
ton
Glo
ba
l R
ou
tin
g
�O
bje
ctives
�M
inim
ize w
ire le
ngth
�B
ala
nce c
ongestion
�T
imin
g d
riven �
main
tain
tim
ing c
onstr
ain
ts
�N
ois
e d
rive
n �
min
imiz
e c
ross-c
oup
led c
apa
citance
�K
ee
p b
use
s togeth
er
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
5
Kah
ng
/Keu
tzer/
New
ton
Glo
ba
l R
ou
tin
g F
orm
ula
tio
n
Giv
en
(i)
Pla
cem
en
t o
f b
locks/c
ell
s
(ii)
ch
an
nel
cap
acit
ies
Dete
rmin
eR
ou
tin
g t
op
olo
gy o
f each
net
Op
tim
ize
(i)
max #
nets
ro
ute
d(i
i) m
in r
ou
tin
g a
rea
(iii
) m
in t
ota
l w
irele
ng
th
Cla
ssic
te
rmin
olo
gy:
In
g
en
era
l cell
d
esig
n
or
sta
nd
ard
cell
desig
n,
we a
re a
ble
to
mo
ve b
locks
or
cell
ro
ws,
so
we c
an
gu
ara
nte
e c
on
necti
on
s o
f all
th
e n
ets
(“va
riab
le-d
ie”
+ c
han
nel ro
ute
rs).
Cla
ssic
te
rmin
olo
gy:
In
g
ate
-arr
ay
desig
n,
exce
ed
ing
ch
an
nel
cap
acit
y i
s n
ot
all
ow
ed
(“fi
xed
-
die
”+
are
a r
ou
ters
).
Sin
ce
Tan
gen
t’s
Tan
cell
(~1986),
an
d
>
3L
M
pro
cesse
s,
we
use
larg
ely
u
se
are
a
rou
ters
fo
r cell
-based
layo
ut
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
6
Kah
ng
/Keu
tzer/
New
ton
Glo
ba
l R
ou
tin
g
�P
rovid
e g
uid
ance t
o d
eta
iled r
outin
g (
why?)
�O
bje
ctive functio
n is a
pp
licatio
n-d
ep
en
de
nt
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
7
Kah
ng
/Keu
tzer/
New
ton
Gra
ph
Mo
de
ls f
or
Glo
ba
l R
ou
tin
g
�G
loba
l ro
utin
g p
rob
lem
is a
gra
ph p
roble
m
�M
ode
l ro
uting r
egio
ns,
their a
dja
cencie
s a
nd
capacitie
s
as g
raph v
ert
ices,
edges a
nd w
eig
hts
�C
hoic
e o
f m
ode
l dep
ends o
n a
lgorith
m
�G
rid g
raph m
ode
l
�G
rid g
raph
repre
sents
layout as a
hXw
arr
ay, vert
ices a
re layout
cells
, edges c
aptu
re c
ell
adja
cencie
s, zero
-capacity e
dges
repre
sent blo
cked c
ells
�C
han
ne
l in
ters
ection g
raph m
ode
l fo
r blo
ck-b
ased
desig
n
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
8
Kah
ng
/Keu
tzer/
New
ton
Ch
an
ne
l In
ters
ec
tio
n G
rap
h
�E
dg
es a
re c
han
ne
ls,
vert
ices a
re c
han
ne
l in
ters
ections
(CI)
, v1 a
nd v
2 a
re a
dja
cent
if there
exis
ts a
chan
ne
l betw
een (
CI 1
and C
I 2).
Gra
ph c
an b
e e
xte
nded t
o inclu
de
pin
s.
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
9
Kah
ng
/Keu
tzer/
New
ton
Glo
ba
l R
ou
tin
g A
pp
roa
ch
es
�C
an r
oute
nets
:
�S
equentially
, e.g
. one a
t a tim
e –
e.g
. usin
g m
aze r
outing b
ut
just assig
n to a
routing r
esourc
e
�C
oncurr
ently, e.g
. sim
ultaneously
all
nets
�S
equential appro
aches
�S
ensitiv
e to o
rdering
�U
sually
sequenced b
y
-C
riticalit
y
-N
um
ber
of te
rmin
als
�C
oncurr
ent appro
aches
�C
om
puta
tionally
hard
�H
iera
rchic
al m
eth
ods u
sed
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 2
0
Kah
ng
/Keu
tzer/
New
ton
Ta
xo
no
my o
f V
LS
I R
ou
ters
Gra
ph
Searc
h
Ste
iner
Itera
tiv
e
Hie
rarc
hic
al
Gre
ed
yL
eft
-Ed
ge
Riv
er
Sw
itch
bo
x
Ch
an
nel
Maze
Lin
e P
rob
e
Lin
e E
xp
ansi
on
Res
tric
ted
Gen
era
l P
urp
ose
Po
wer
& G
rou
nd
Clo
ck
Glo
bal
Deta
iled
Sp
ecia
lize
d
Ro
ute
rs
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 2
1
Kah
ng
/Keu
tzer/
New
ton
Clo
ck
Ro
uti
ng
Str
uc
ture
s
Ba
lan
ce
d T
ree
H-T
ree
Or
in lo
w p
erf
orm
an
ce A
SIC
desig
ns a
clo
ck m
ay b
e
“ju
st
an
oth
er
net”
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 2
2
Kah
ng
/Keu
tzer/
New
ton
Clo
ck
Ro
uti
ng
Tru
nk
or
Gri
dT
run
k o
r G
rid
�M
ult
iple
Clo
ck
Do
ma
ins
�M
ult
iple
Clo
ck
Do
ma
ins
Clo
ck
Me
sh
Clo
ck
Me
sh
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 2
3
Kah
ng
/Keu
tzer/
New
ton
Po
we
r R
ou
tin
g
So
urc
eS
ou
rce
�P
ow
er
Mesh
�P
ow
er
Rin
g
�S
tar
Routing
�P
ow
er
Mesh
�P
ow
er
Rin
g
�S
tar
Routing
Sta
r R
outin
gS
tar
Routin
g
In c
ases w
here
an
en
tire
layer
is d
evo
ted
to
po
wer,
bo
th N
/S,
E/W
dir
ecti
on
s m
ay b
e u
sed
.
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 2
4
Kah
ng
/Keu
tzer/
New
ton
Ta
xo
no
my o
f V
LS
I R
ou
ters
Gra
ph
Searc
h
Ste
iner
Itera
tiv
e
Hie
rarc
hic
al
Gre
ed
yL
eft
-Ed
ge
Riv
er
Sw
itch
bo
x
Ch
an
nel
Maze
Lin
e P
rob
e
Lin
e E
xp
ansi
on
Res
tric
ted
Gen
era
l P
urp
ose
Po
wer
& G
rou
nd
Clo
ck
Glo
bal
Deta
iled
Sp
ecia
lize
d
Ro
ute
rs
X {
grid
ded,
grid
less
}
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 2
5
Kah
ng
/Keu
tzer/
New
ton
On
e L
aye
r R
ou
tin
g:
Ge
ne
ral
Riv
er-
Ro
uti
ng
�F
or
clo
ck, pow
er,
gro
und s
till
may n
eed to s
olv
e s
ingle
-layer
routing
�T
wo p
ossib
le p
ath
s p
er
net alo
ng b
oundary
�P
ath
= a
ltern
ating s
equ
ence o
f horizonta
l a
nd v
ert
ical segm
ents
connecting t
wo
term
inals
of
a n
et
�C
onsid
er
sta
rtin
g term
inals
and e
ndin
g term
inals
�A
ssum
e e
very
path
counte
r-clo
ckw
ise a
round b
oundary
T1
T2P1
P2
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 2
6
Kah
ng
/Keu
tzer/
New
ton
On
e L
aye
r R
ou
tin
g:
Ge
ne
ral
Riv
er-
Ro
uti
ng
�C
reate
circula
r lis
t of all
term
inals
ord
ere
d
counte
rclo
ckw
ise a
ccord
ing t
o p
ositio
n o
n b
ou
nd
ary
1s
1e
3e
8e7e
6e5e
4e
2e
2s3s
4s5s
6s
7s
8s
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 2
7
Kah
ng
/Keu
tzer/
New
ton
�B
ou
nd
ary
-packe
d s
olu
tion
�F
lip c
orn
ers
to m
inim
ize w
ire len
gth
1s
1e
3e
8e7e
6e5e
4e
2e
2s3s
4s5s
6s
7s
8s
On
e L
aye
r R
ou
tin
g:
Ge
ne
ral
Riv
er-
Ro
uti
ng
On
e L
aye
r R
ou
tin
g:
Ge
ne
ral
Riv
er-
Ro
uti
ng
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 2
8
Kah
ng
/Keu
tzer/
New
ton
Ta
xo
no
my o
f V
LS
I R
ou
ters
Gra
ph
Searc
h
Ste
iner
Itera
tiv
e
Hie
rarc
hic
al
Gre
ed
yL
eft
-Ed
ge
Riv
er
Sw
itch
bo
x
Ch
an
nel
Maze
Lin
e P
rob
e
Lin
e E
xp
ansi
on
Res
tric
ted
Gen
era
l P
urp
ose
Po
wer
& G
rou
nd
Clo
ck
Glo
bal
Deta
iled
Sp
ecia
lize
d
Ro
ute
rs
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 2
9
Kah
ng
/Keu
tzer/
New
ton
Ro
uti
ng
La
ye
r M
od
els
HV
mode
lH
V m
ode
lV
H m
ode
lV
H m
ode
l
HV
H m
ode
lH
VH
mode
lV
HV
mod
el
VH
V m
od
el
Layer
1Layer
1
Layer
2Layer
2
Layer
3Layer
3
Via
Via
1 layer
1 layer
2 layers
2 layers
3 layers
3 layers
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 3
0
Kah
ng
/Keu
tzer/
New
ton
Ch
an
ne
l v
s.
Sw
itc
hb
ox
Ch
an
nel
�C
han
nel m
ay h
ave e
xit
s a
t le
ft a
nd
ri
gh
t sid
es, b
ut
exit
po
sit
ion
s a
re
no
t fi
xed
�W
e m
ay m
ap
exit
s t
o e
ith
er
low
er
or
up
per
ed
ge o
f a c
han
nel
�O
ne d
imen
sio
nal p
rob
lem
Sw
itch
bo
x
�T
erm
inal p
osit
ion
s o
n a
ll f
ou
r sid
es o
f a s
wit
ch
bo
x a
re f
ixed
�T
wo
dim
en
sio
nal p
rob
lem
Sw
itch
bo
x r
ou
tin
g is m
ore
dif
ficu
lt
1 11
32
44
4
43
32
2
3
3
3
1
1
12
22
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 3
1
Kah
ng
/Keu
tzer/
New
ton
Ch
an
ne
l R
ou
tin
g P
rob
lem
Inp
ut:
Pin
s o
n t
he lo
wer
an
d u
pp
er
ed
ge
Ou
tpu
t: C
on
necti
on
of
each
net
Co
nstr
ain
ts (
Assu
mp
tio
n)
(i)
g
rid
str
uctu
re(i
i) tw
o r
ou
tin
g la
yers
. O
ne f
or
ho
rizo
nta
l w
ires, th
e o
ther
for
vert
ical w
ires
(iii)
via
sfo
r co
nn
ecti
ng
wir
es in
tw
o la
yers
Min
imiz
e:
(i)
# t
racks (
ch
an
nel h
eig
ht)
(ii)
to
tal w
ire len
gth
(iii)
# v
ias
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 3
2
Kah
ng
/Keu
tzer/
New
ton
Ch
an
ne
l R
ou
tin
g
�B
asic
Term
inolo
gy:
�F
ixed p
in p
ositio
ns o
n top a
nd b
ottom
edges
�C
lassic
al channel: n
o n
ets
leave c
hannel
�T
hre
e-s
ided c
hannel possib
le
chan
nel
pin
s
trac
ksen
d
via
tru
nk
net
11
1
bra
nch
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 3
3
Kah
ng
/Keu
tzer/
New
ton
Ch
an
nel/
Sw
itch
Bo
x R
ou
tin
g A
lgo
rith
ms
�G
rap
h t
heo
ry b
ased
alg
ori
thm
Yo
sh
imu
ra a
nd
Ku
h
�G
reed
y a
lgo
rith
mR
ivest
an
d F
idu
ccia
�M
aze r
ou
tin
g a
nd
its
vari
ati
on
sL
ee, R
ob
in, S
ou
ku
p, O
hts
uki
�H
iera
rch
ical w
ire r
ou
tin
gB
urs
tein
an
d P
ela
vin
Ch
an
nel ro
uti
ng
Ch
an
nel
/ sw
itch
bo
x
an
d
gen
era
l are
a
rou
tin
g
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 3
4
Kah
ng
/Keu
tzer/
New
ton
Gre
ed
y C
ha
nn
el
Ro
ute
r
R.L
. R
ivest
an
d
C.M
. F
idu
ccia
“A
G
reed
y
Ch
an
nel
Ro
ute
r”, 19th
DA
C, 1982 P
418-4
24
oA
sim
ple
lin
ear
tim
e a
lgo
rith
m
oG
uara
nte
e t
he c
om
ple
tio
n o
f all t
he n
ets
(m
ay e
xte
nd
to
rig
ht-
han
d s
ide o
f th
e c
han
nel)
oP
rod
uce
bo
th
restr
icte
d
do
gle
gs
an
d
un
restr
icte
d
do
gle
gs
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 3
5
Kah
ng
/Keu
tzer/
New
ton
Gre
ed
y R
ou
ter:
Riv
est
& F
idu
ccia
�P
roceed c
olu
mn b
y c
olu
mn (
left to r
ight)
�M
ake c
onnections to a
ll pin
s in that colu
mn
�F
ree u
p tra
cks b
y c
olla
psin
g a
s m
any tra
cks a
s p
ossib
le to
colla
pse n
ets
�S
hrink r
ange o
f ro
ws o
ccupie
d b
y a
net by u
sin
g d
ogle
gs
�If a
pin
cannot ente
r a c
hannel, a
dd a
tra
ck
�O
(pin
s)
tim
e
14
61
7
8
49
1010
32
5
26
98
79
35
5
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 3
6
Kah
ng
/Keu
tzer/
New
ton
Left
-to
-rig
ht,
Co
lum
n-b
y-c
olu
mn
scan
c:=
0;
wh
ile (
no
t d
on
e)
do
beg
in c:=
c+
1;
co
mp
lete
wir
ing
at
co
lum
n c
;en
d;
In g
en
era
l, a
t a p
oin
t in
a n
et
ma
y b
e
(1)
em
pty
(n
et
5)
(2)
un
sp
lit
(nets
1,4
)
(3)
sp
lit
(net
3)
(4)
co
mp
lete
d (
net2
)
Ov
erv
iew
of
Gre
ed
y R
ou
ter
1
3
1
2
1
5
2
1
2
3
4
5
4
4
1 3
4
3
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 3
7
Kah
ng
/Keu
tzer/
New
ton
Pa
ram
ete
rs t
o G
ree
dy R
ou
ter
•In
itia
l-ch
an
nel-
wid
th icw
•M
inim
um
-jo
g-l
en
gth
m
jl(W
hy?
)
•S
tead
y-n
et-
co
nsta
nt
sn
c
•U
su
ally s
tart
icw
as d
. th
e d
en
sit
y
•M
jlco
ntr
ols
th
e n
um
ber
of
via
s, u
se a
larg
e
mjl
for
few
er
via
s
•S
nc
als
o c
on
tro
ls #
of
via
s(t
yp
ical valu
e=
10
Wh
y?
)
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 3
8
Kah
ng
/Keu
tzer/
New
ton
Op
era
tio
ns
at
Ea
ch
Co
lum
n
At
each
co
lum
n, th
e g
reed
y r
ou
ter
trie
s t
o m
axim
ize
the u
tility
of
the w
irin
g p
rod
uced
:
A:
Make m
inim
al fe
asib
le t
op
/bo
tto
m c
on
necti
on
s;
B:
Co
llap
se (
co
nn
ect)
sp
lit
nets
;
C:
Mo
ve s
plit
nets
clo
ser
to o
ne a
no
ther;
D:
Rais
e r
isin
g n
ets
/lo
wer
fallin
g n
ets
, I.e. b
rin
g n
ets
clo
ser
to d
esti
nati
on
term
inal;
E:
Wid
en
ch
an
nel w
hen
necessary
;
F:
Exte
nd
to
next
co
lum
n;
1
3
1
2
1
5
2
1
2
3
4
5
4
4
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 3
9
Kah
ng
/Keu
tzer/
New
ton
(A)
Make M
inim
al F
easib
le t
op
/bo
tto
m C
on
neti
on
s
2 3 4
2 3 4
44
1
1
2 3 1
2 3 1
00
1
1
A
A*
B
B*
2 12 1
22
1
1
C
C*
1 2 3 4
50
D
D*
1 2 3 4
50
1 2 3 4
55 F
F
*
1 2 3 4
55
3 2 1 4
32
E
E*
3 2 1 4
32
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 4
0
Kah
ng
/Keu
tzer/
New
ton
( B
) C
oll
ap
se
/(C
on
ne
ct)
Sp
lit
Ne
ts
0
2
1 3 2
0
2
1 3 2
G
G*
0 0
1 3 4 1 4
0
H
H
*
0
1 3 4 1 4
0
1
2 1 2 3 4 3
I
I*0
1
2 1 2 3 4 3
0
0
1 2 1 20
1 2 1 2
J
J
*
0
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 4
1
Kah
ng
/Keu
tzer/
New
ton
( C
) M
ov
e S
pli
t N
ets
Clo
se
r
0
0
1 2 1 2
k
k*
0
0
1 2 1 2
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 4
2
Kah
ng
/Keu
tzer/
New
ton
L
L*
0
1
3 4
fall
ing
fall
ing
risin
g
0
1
3 4
fall
ing
fall
ing
risin
g
(D)
Ris
ing
/Fa
llin
g
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 4
3
Kah
ng
/Keu
tzer/
New
ton
M
M
*
70
1 5 1 6
70
1 5 1 6
(E)
Ins
ert
Ne
w T
rac
k
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 4
4
Kah
ng
/Keu
tzer/
New
ton
70
1 5 1 6
7
N
(F)
Ex
ten
d t
o N
ex
t C
olu
mn
5
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 4
5
Kah
ng
/Keu
tzer/
New
ton
Gre
ed
y R
ou
tin
g E
xa
mp
le
14
61
7
8
49
1010
32
5
26
98
79
35
5
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 4
6
Kah
ng
/Keu
tzer/
New
ton
Co
mm
en
ts o
n G
ree
dy R
ou
ter
(Riv
est&
Fid
uccia
1982)
•A
lways s
ucce
ed
s (
even
if
cycli
c c
on
flic
t is
pre
sen
t);
•A
llo
ws u
nre
str
icte
d d
og
leg
s;
•A
llo
ws
a n
et
to o
ccu
py m
ore
th
an
1 t
rack a
t a g
iven
co
lum
n;
•M
ay u
se a
few
co
lum
ns o
ff t
he e
dg
e;
1 3
1
2
1 5
2
1
3
3
4
5 4
4
A
co
lum
n
off
th
e e
dg
eUn
restr
icte
d
do
gle
g
Net
1 o
ccu
pie
s t
wo
tra
ck a
t th
is
co
lum
n;
it a
lso
“w
rap
s-a
rou
nd
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 4
7
Kah
ng
/Keu
tzer/
New
ton
Wh
at’
s w
ron
g w
ith
ch
an
ne
l ro
uti
ng
?
�W
e n
eed
to b
e a
ble
to
route
over
cells
an
d
get
to p
ins
�W
e w
ant
to m
inim
ize
routing a
rea
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 4
8
Kah
ng
/Keu
tzer/
New
ton
Ta
xo
no
my o
f V
LS
I R
ou
ters
Gra
ph
Searc
h
Ste
iner
Itera
tiv
e
Hie
rarc
hic
al
Gre
ed
yL
eft
-Ed
ge
Riv
er
Sw
itch
bo
x
Ch
an
nel
Maze
Lin
e P
rob
e
Lin
e E
xp
ansi
on
Res
tric
ted
Gen
era
l P
urp
ose
Po
wer
& G
rou
nd
Clo
ck
Glo
bal
Deta
iled
Sp
ecia
lize
d
Ro
ute
rs
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 4
9
Kah
ng
/Keu
tzer/
New
ton
Tw
o-T
erm
ina
l R
ou
tin
g:
Ma
ze
Ro
uti
ng
�M
aze r
outing f
inds a
path
betw
een s
ourc
e (
s)
and t
arg
et
(t)
in a
pla
nar
gra
ph o
n a
fix
ed g
rid
�G
rid g
raph m
ode
l is
used t
o r
epre
sent
blo
ck p
lacem
ent
�A
va
ilab
le r
outin
g a
rea
s a
re u
nblo
cked v
ert
ices,
obsta
cle
s
are
blo
cke
d v
ert
ices
�F
inds a
n o
ptim
al path C
ou
rtesy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 5
0
Kah
ng
/Keu
tzer/
New
ton
Ma
ze
Ro
uti
ng
4 3
2 3 4 5 6
7 8
9
10 1
1
3 2
1 2 3
4 5 6 7
8
9 1
0
2 1 A
1
5
6 7
8
3 2
1
2 6 7
8
9
10 11 12
4 3
2
3
12 13
5 4
3
4 1
4 B
13 14
6 5
1
3 1
4
14
7 6
7
11 1
2 13 14
8 7 8 9 1
0 11 1
2 13 14
9 8 9 1
0 11 1
2 13 14
�P
oin
t to
poin
t ro
uting o
f nets
�R
oute
fro
m s
ourc
e to s
ink
�B
asic
idea =
wave p
ropagation (
Lee, 1961)
�B
readth
-first searc
h +
back-t
racin
g a
fter
findin
g s
hort
est path
�G
uara
nte
es to fin
d the s
hort
est path
�O
bje
ctive =
route
all
nets
accord
ing to s
om
e c
ost fu
nction that
min
imiz
es c
ongestion, ro
ute
length
, couplin
g, etc
.
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
“A
n A
lgo
rith
m
for
Path
C
on
necti
on
an
d
its
Ap
plica
tio
n”,
C.Y
. L
ee
, IR
E
Tra
nsacti
on
s o
n
Ele
ctr
on
ic
Co
mp
ute
rs, 196
1.
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 5
1
Kah
ng
/Keu
tzer/
New
ton
Ma
ze
Ro
uti
ng
S
TE
. Y
ou
ng
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 5
2
Kah
ng
/Keu
tzer/
New
ton
Ba
sic
Id
ea
�A
Bre
adth
-First
Searc
h(B
FS
) of th
e g
rid g
raph.
�A
lways f
ind
the s
hort
est path
possib
le.
�C
onsis
ts o
f tw
o p
hase
s:
�W
ave p
rop
agation
�R
etr
ace
E. Y
ou
ng
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 5
3
Kah
ng
/Keu
tzer/
New
ton
An
Ill
us
tra
tio
n
S
T
01
1
2
2
4
46
3
3
3 5
55
E. Y
ou
ng
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 5
4
Kah
ng
/Keu
tzer/
New
ton
Wa
ve
Pro
pa
ga
tio
n
�A
t ste
p k
, all
vert
ices a
t M
anhatt
an-d
ista
nce k
from
Sare
lab
ele
d w
ith k
.
�A
Pro
pag
atio
n L
ist
(FIF
O)
is u
sed t
o k
eep tra
ck o
f th
e
vert
ices t
o b
e c
onsid
ere
d n
ext.
S
T
0S
T
01
2
12 3
45
45
6
3
3S
T
01
2
12 3
3
3
5
Aft
er S
tep 0
Aft
er S
tep 3
Aft
er S
tep 6
E. Y
ou
ng
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 5
5
Kah
ng
/Keu
tzer/
New
ton
Re
tra
ce
�T
race b
ack the a
ctu
al ro
ute
.
�S
tart
ing f
rom
T.
�A
t vert
ex w
ith k
, go to a
ny v
ert
ex w
ith lab
elk-1
.
S
T
01
2
12 3
45
45
6
3
3
5 Fin
al l
abel
ing
E. Y
ou
ng
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 5
6
Kah
ng
/Keu
tzer/
New
ton
Ho
w m
an
y g
rid
s v
isit
ed
usin
g L
ee
’s a
lgo
rith
m?
S
T
11
11
22
222
2 333
33
33
344
4
44
44
55
55
5
55
556
66
66
6
66
66
66
66
777
7
77
7 77
77
77
77
778
88
88
88
88
88
8
889
99
99
99
99
9
99
999
99
910
1010 1
0
10
10
10
10
10
10
10
10
10
10
10
10
101
010
11111
1
1111
1111
11
1111 1
1111
1111
1
11
1112
12
12
12
12
12
12
12
12
12
1212
12121
212
13
13
13
13
13
13
13
13
13
13
13
13
13
E. Y
ou
ng
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 5
7
Kah
ng
/Keu
tzer/
New
ton
Tim
e a
nd
Sp
ac
e C
om
ple
xit
y
�F
or
a g
rid s
tructu
re o
f siz
e w
×h:
•T
ime p
er
net =
O(w
h)
•S
pace =
O(w
hlo
g w
h)
(O
(log
wh)
bits a
re n
eeded to s
tore
each label.)
�F
or
a 4
000 ×
4000 g
rid
str
uctu
re:
•24 b
its p
er
label
•T
ota
l 48 M
byte
s o
f m
em
ory
!
�G
enera
lly,
tim
e a
nd s
pace c
om
ple
xity p
er
net
O(h
eig
ht
Xw
idth
)
E. Y
ou
ng
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 5
8
Kah
ng
/Keu
tzer/
New
ton
Le
e’s
Alg
ori
thm
�S
trength
s:
�G
uara
nte
e to fin
d a
connection if one e
xis
ts.
�G
uara
nte
e s
hort
est path
.
�W
eaknesses:
�Larg
e m
em
ory
requirem
ents
.
�S
low
�A
pp
lication
s:
�G
lobal ro
uting a
nd d
eta
iled r
outing
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 5
9
Kah
ng
/Keu
tzer/
New
ton
Imp
rov
em
en
t to
Le
e’s
Alg
ori
thm
�Im
pro
vem
ent
on m
em
ory
:
�A
ker’
sC
odin
g S
chem
e
�Im
pro
vem
ent
on r
un tim
e:
�S
tart
ing p
oin
t sele
ction
�D
ouble
fan-o
ut
�F
ram
ing
�H
adlo
ck’s
Alg
orith
m
�S
oukup
’sA
lgorith
m
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 6
0
Kah
ng
/Keu
tzer/
New
ton
Lim
itin
g t
he
Se
arc
h R
eg
ion
�S
ince m
ajo
rity
of nets
are
route
d w
ithin
the b
oun
din
g b
ox
defin
ed b
y S
and T
, can lim
it p
oin
ts s
earc
hed
by m
aze
route
r to
those w
ithin
bou
nd
ing b
ox
�A
llow
s m
aze r
oute
r to
fin
ish s
ooner
with little o
r no n
egative
impact on fin
al ro
uting c
ost
�R
oute
r w
ill n
ot consid
er
poin
ts that are
unlik
ely
to b
e o
n the
route
path
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 6
1
Kah
ng
/Keu
tzer/
New
ton
Pro
ble
ms
Wit
h M
aze
Ro
uti
ng
Slo
w:
for
each
net,
we h
ave t
o s
earc
h N
× ×××N
gri
d
Mem
ory
: to
tal la
yo
ut
gri
d n
eed
s t
o b
e k
ep
t N
xN
Imp
rovem
en
ts�
Sim
ple
sp
eed
-up
�M
inim
um
deto
ur
alg
ori
thm
(H
ad
lock, 1977)
�F
ast
maze a
lgo
rith
m (
So
uku
p, 1978)
�d
ep
th-f
irst
searc
h u
nti
l o
bsta
cle
�b
read
th-f
irst
at
ob
sta
cle
�u
nti
l ta
rget
is r
each
ed
�W
ill fi
nd
a p
ath
if
it e
xis
ts, m
ay b
e s
ub
op
tim
al
�T
yp
ical sp
eed
-up
10-5
0x
Fu
rth
er
imp
rovem
en
ts�
Maze r
ou
tin
g in
feasib
le f
or
larg
e c
hip
s�
Lin
e s
earc
h (
Mik
am
i&
Tab
uch
i, 1
968;
Hig
hto
wer,
1969)
�P
att
ern
ro
uti
ng
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 6
2
Kah
ng
/Keu
tzer/
New
ton
Gri
d P
rob
lem
wit
h M
aze
Ro
uti
ng �
Wh
at’
s t
he g
rid
?
�E
as
y i
f:
�G
lob
al ro
ute
r d
oes l
aye
r assig
nm
en
t o
r
�T
SM
C s
tyle
-sets
la
ye
r w
idth
th
e s
am
e f
or
layers
1-6
(8)
�H
ard
if
maze r
ou
ter
pre
su
med
to
be
mu
ltil
ayer
an
d
dif
fere
nt
wir
e w
idth
s
for
each
layer:
�M
1
105
�M
2
105
�M
3
110
�M
4
140
�M
5
165
�M
6
240
�M
7
360
�M
8
540
Ph
oto
co
urt
es
y:
Ja
n M
. R
ab
ae
yA
na
nth
aC
ha
nd
rakasa
nB
ori
vo
je N
iko
lic
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 6
3
Kah
ng
/Keu
tzer/
New
ton
Lin
e-P
rob
e A
lgo
rith
m
Mik
am
i&T
ab
uch
iIF
IPS
Pro
c,
Vo
lH
47,
pp
1475-1
47
8,
19
68
Mik
am
i+T
ab
uch
i’s
alg
ori
thm
�G
en
era
te s
earc
h lin
es f
rom
bo
th s
ou
rce a
nd
targ
et
(level-
0
lin
es)
�F
rom
every
po
int
on
th
e level-
i searc
h lin
es, g
en
era
te
perp
en
dic
ula
r le
vel-
(i+
1)
searc
h lin
es
�P
roceed
un
til a s
earc
h lin
e f
rom
th
e s
ou
rce m
eets
a
searc
h lin
e f
rom
a t
arg
et
�W
ill fi
nd
th
e p
ath
if
it e
xis
ts, b
ut
no
t g
uara
nte
ed
to
fin
d t
he
sh
ort
est
path
Tim
e a
nd
sp
ace c
om
ple
xit
y:
O(L
), w
here
Lis
th
e n
um
ber
of
lin
e s
eg
men
ts
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 6
4
Kah
ng
/Keu
tzer/
New
ton
Lin
e-P
rob
e S
um
ma
ry
�F
as
t, h
an
dle
s la
rge
ne
ts / d
ista
nc
es
/ d
es
ign
s
�R
ou
tin
g m
ay b
e in
co
mp
lete
Targ
et
pro
be
Inte
rsection o
f escape lin
e
Escape lin
e
Sourc
e p
robe
Escape lin
e
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 6
5
Kah
ng
/Keu
tzer/
New
ton
Pro
ble
ms
wit
h S
eq
ue
nti
al
Ro
uti
ng
Alg
ori
thm
s
Net
ord
eri
ng
�M
ust
rou
te n
et
by n
et,
bu
t d
iffi
cu
lt t
o d
ete
rmin
e b
est
net
ord
eri
ng
!
�D
iffi
cu
lt t
o p
red
ict/
avo
id c
on
gesti
on
Wh
at
can
be d
on
e
�U
se o
ther
rou
ters
�C
han
nel/
sw
itch
bo
x r
ou
ters
�H
iera
rch
ical
rou
ters
�R
ip-u
p a
nd
rero
ute
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 6
6
Kah
ng
/Keu
tzer/
New
ton
A P
otp
ou
rri
of
Oth
er
Ro
uti
ng
Is
su
es
�M
ore
than a
ny o
ther
top
ic t
hat w
e w
ill s
tudy, ro
uting a
nd
com
pactio
n f
ace a
larg
e n
um
ber
of
issues a
risin
g f
rom
id
iosyncra
sie
s o
f sem
icond
ucto
r pro
cesses
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 6
7
Kah
ng
/Keu
tzer/
New
ton
De
tail
ed
Ro
uti
ng
Is
su
es
�R
outing c
om
ple
tion
�W
idth
and s
pacin
g r
ule
�M
inim
um
wid
th a
nd
sp
acin
g
�V
ari
ab
le w
idth
an
d s
pa
cin
g
-C
on
necti
on
-N
et
-C
lass o
f n
ets
�T
ap
eri
ng
Po
ly
Tap
eri
ng
M2
M2
M1
M1
M1
M1
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 6
8
Kah
ng
/Keu
tzer/
New
ton
De
tail
ed
Ro
uti
ng
Is
su
es 0.4
m0.4
m
0.8
m0.8
m
>=
2m
>=
2m
>=
2m
>=
2m
0.6
m0.6
m
>=
2m
>=
2m
Wid
th-b
ased
Sp
acin
g
�W
idth
and s
pacin
g r
ule
�W
idth
and s
pacin
g r
ule
Min
imu
m
sp
acin
g
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 6
9
Kah
ng
/Keu
tzer/
New
ton
De
tail
ed
Ro
uti
ng
Is
su
es
Spacing
Extra space
Segregation
Noisy region
Quiet region
Shielding
Grounded Shields
�N
ois
e-d
riven –
i.e. nois
e m
inim
ization d
riven
�N
ois
e-d
riven –
i.e. nois
e m
inim
ization d
riven
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 7
0
Kah
ng
/Keu
tzer/
New
ton
De
tail
ed
Ro
uti
ng
Is
su
es
Po
wer
Po
wer
Sig
nal
Sig
nal
Gro
un
dG
rou
nd
Sam
e-L
ayer
Sh
ield
ing
M2
M2
M1
M1
Po
lyP
oly
Sig
nal
Sig
nal
Ad
jacen
t-L
ayer
Sh
ield
ing
�S
hie
ldin
g�S
hie
ldin
g
�S
am
e-l
aye
r shie
ldin
g�S
am
e-l
aye
r shie
ldin
g
�A
dja
ce
nt-
layer
shie
ldin
g�A
dja
ce
nt-
layer
shie
ldin
g
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 7
1
Kah
ng
/Keu
tzer/
New
ton
De
tail
ed
Ro
uti
ng
Is
su
es
Sh
ield
Sh
ield
Bu
sB
us
Bu
s S
hie
ldin
g
�S
hie
ldin
g�S
hie
ldin
g
�B
us s
hie
ldin
g�B
us s
hie
ldin
g
�B
us inte
rle
avin
g�B
us inte
rle
avin
g
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 7
2
Kah
ng
/Keu
tzer/
New
ton
De
tail
ed
Ro
uti
ng
Is
su
es
Dif
fere
nti
al
Dif
fere
nti
al
Ba
lan
ce
d l
en
gth
Ba
lan
ce
d l
en
gth
�D
iffe
ren
tia
l p
air
ro
utin
g
�B
ala
nce
d le
ng
th o
r ca
pa
citan
ce
�D
iffe
ren
tia
l p
air
ro
utin
g
�B
ala
nce
d le
ng
th o
r ca
pa
citan
ce
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 7
3
Kah
ng
/Keu
tzer/
New
ton
De
tail
ed
Ro
uti
ng
Is
su
es
�B
us R
ou
tin
g�
Bu
s R
ou
tin
g
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 7
4
Kah
ng
/Keu
tzer/
New
ton
Cu
rre
nt
Sta
tus
�R
ou
tin
g 1
00,0
00 n
ets
is r
ou
tin
e
�F
un
dam
en
tal p
rob
lem
s
�T
imin
g d
riven
ro
uti
ng
�T
acti
cal
pro
ble
ms
�In
cre
asin
g c
om
ple
xit
y/r
estr
icti
on
s o
f d
esig
n r
ule
s
�R
ou
tin
g e
ven
on
e n
et
DR
C (
desig
n r
ule
ch
ecked
) co
rrec
tly i
s a
ch
all
en
ge (
Jaso
n C
on
g p
ap
er)
�R
ou
tin
g i
s n
ot
a “
sh
ow
sto
pp
er”
in c
urr
en
t V
LS
I d
esig
ns
�B
ut
…ro
uti
ng
is a
pere
nn
ial b
ott
len
eck,
an
d t
here
fore
th
ere
’s a
t le
ast
on
e n
ew
sta
rt-u
p e
very
yea
r
�F
ocu
s o
f sta
rt-u
ps i
s:
�C
hip
-le
vel
rou
tin
g f
or
asse
mb
ly
�S
pecia
l p
urp
ose n
ich
e r
ou
tin
g –
e.g
. d
ie t
o b
on
din
g p
ad
s t
o
packag
e
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 7
5
Kah
ng
/Keu
tzer/
New
ton
Ro
uti
ng
Su
mm
ary
Gra
ph
Searc
h
Ste
iner
Itera
tiv
e
Hie
rarc
hic
al
Gre
ed
yL
eft
-Ed
ge
Riv
er
Sw
itch
bo
x
Ch
an
nel
Maze
Lin
e P
rob
e
Lin
e E
xp
ansi
on
Res
tric
ted
Gen
era
l P
urp
ose
Po
wer
& G
rou
nd
Clo
ck
Glo
bal
Deta
iled
Sp
ecia
lize
d
Ro
ute
rs
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
Ro
uti
ng
req
uir
es a
n e
sp
ecia
lly l
arg
e t
oo
lbo
x o
f te
ch
niq
ues
Eve
ry r
ou
ter
men
tio
ned
here
is u
sed
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 7
6
Kah
ng
/Keu
tzer/
New
ton
Su
mm
ary
�Larg
e t
oo
lset �
many a
lgori
thm
s
�F
ort
unate
ly, m
ost of th
e a
lgorith
ms a
re s
imple
�B
e s
ure
you’re f
am
ilia
r w
ith:
�M
aze R
outing –
Lee
�Lin
e r
outing -
Hig
hto
wer
�S
ingle
-layer
sw
itchbox r
outing
�G
reedy c
hannel ro
uting -
Riv
est
�B
e f
am
iliar
with t
he flo
w a
nd r
ela
tio
nsh
ip b
etw
een r
oute
rs
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 7
7
Kah
ng
/Keu
tzer/
New
ton
Ex
tra
s
�A
ker’s
Cod
ing S
chem
e f
or
Maze R
outin
g
�W
hat’s t
here
to d
o in c
han
ne
l ro
utin
g?
�P
att
ern
-based r
outin
g
�S
tein
er
Tre
es
�C
oncurr
ent
glo
ba
l ro
utin
g
�Y
osh
imura
-Kuh
�C
om
pactio
n
�O
ther
Physic
al Is
sues
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 7
8
Kah
ng
/Keu
tzer/
New
ton
Ak
er’
sC
od
ing
Sc
he
me
�F
or
the L
ee
’s a
lgorith
m, la
be
ls a
re n
eed
ed d
uri
ng
the r
etr
ace p
hase.
�B
ut th
ere
are
only
tw
o p
ossib
le la
be
ls f
or
the
ne
ighb
ors
of
each v
ert
ex la
be
led
i, w
hic
h a
re,
i-1 a
nd
i+1.
�W
ant
a labelin
g s
che
me s
uch t
hat
each lab
el has its
pre
ced
ing labe
l d
iffe
rent fr
om
its
succee
din
g labe
l.
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 7
9
Kah
ng
/Keu
tzer/
New
ton
Ex
tra
s
�W
hat’s t
here
to d
o in c
han
ne
l ro
utin
g?
�P
att
ern
-based r
outin
g
�S
tein
er
Tre
es
�C
oncurr
ent
glo
ba
l ro
utin
g
�Y
osh
imura
-Kuh
�C
om
pactio
n
�O
ther
Physic
al Is
sues
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 8
0
Kah
ng
/Keu
tzer/
New
ton
Tri
via
l C
ha
nn
el
Ro
uti
ng
�A
ssig
n e
ve
ry n
et its o
wn t
rack
Deh
on
-C
alt
ech
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 8
1
Kah
ng
/Keu
tzer/
New
ton
Tri
via
l C
ha
nn
el
Ro
uti
ng
�A
ssig
n e
ve
ry n
et its o
wn t
rack
�C
hannel w
idth
> N
(sin
gle
outp
ut fu
nctions)
�C
hip
bis
ection ∝
N �
chip
are
a N
2
Deh
on
-C
alt
ech
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 8
2
Kah
ng
/Keu
tzer/
New
ton
Sh
ari
ng
Tra
ck
s
�W
ant to
Min
imiz
e t
racks u
sed
�T
rick is t
o s
hare
tra
cks
Deh
on
-C
alt
ech
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 8
3
Kah
ng
/Keu
tzer/
New
ton
No
t th
at
Ea
sy
�W
ith T
wo s
ides
�E
ven a
ssig
nin
g o
ne tra
ck/s
ignal m
ay n
ot be e
nough
A B
B C
C D
Deh
on
-C
alt
ech
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 8
4
Kah
ng
/Keu
tzer/
New
ton
No
t th
at
Ea
sy
�W
ith T
wo s
ides
�E
ven a
ssig
nin
g o
ne tra
ck/s
ignal m
ay n
ot be e
nough
A B
B C
Bad
assig
nm
en
t
Overlap:
A,B
B,C
C
D
Deh
on
-C
alt
ech
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 8
5
Kah
ng
/Keu
tzer/
New
ton
No
t th
at
Ea
sy
�W
ith T
wo s
ides
�E
ven a
ssig
nin
g o
ne tra
ck/s
ignal m
ay n
ot be e
nough
A B
B C
Va
lid a
ssig
nm
ent
avo
ids o
ve
rlap
C
D
Deh
on
-C
alt
ech
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 8
6
Kah
ng
/Keu
tzer/
New
ton
No
t th
at
Ea
sy
�W
ith T
wo s
ides
�E
ven a
ssig
nin
g o
ne tra
ck/s
ignal m
ay n
ot be e
nough
A B
B C
There
are
vert
ical
constr
ain
ts o
n
ord
eri
ng
C D
Deh
on
-C
alt
ech
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 8
7
Kah
ng
/Keu
tzer/
New
ton
Ex
tra
s
�W
hat’s t
here
to d
o in c
han
ne
l ro
utin
g?
�P
att
ern
-based r
outin
g
�S
tein
er
Tre
es
�C
oncurr
ent
glo
ba
l ro
utin
g
�Y
osh
imura
-Kuh
�C
om
pactio
n
�O
ther
Physic
al Is
sues
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 8
8
Kah
ng
/Keu
tzer/
New
ton
Pa
tte
rn-B
as
ed
Ro
uti
ng
�R
estr
ict ro
uting o
f net to
cert
ain
basic
tem
pla
tes
�B
asic
tem
pla
tes a
re L
-shaped (
1 b
end)
or
Z-s
haped (
2 b
ends)
route
s
betw
een a
sourc
e a
nd s
ink
�T
em
pla
tes a
llow
fast ro
uting o
f nets
sin
ce o
nly
cert
ain
edges a
nd
poin
ts a
re c
onsid
ere
d
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 8
9
Kah
ng
/Keu
tzer/
New
ton
Ste
ine
r T
ree
s
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 9
0
Kah
ng
/Keu
tzer/
New
ton
In g
en
era
l, m
aze a
nd
lin
e-p
rob
e r
ou
tin
g a
re n
ot
well-s
uit
ed
to
mu
lti-
term
inal n
ets
Se
vera
l att
em
pts
mad
e t
o e
xte
nd
to
mu
lti-
term
inal n
ets
�C
on
nect
on
e t
erm
inal at
a t
ime
�U
se t
he e
nti
re c
on
necte
d s
ub
trees
as s
ou
rces o
r ta
rgets
d
uri
ng
exp
an
sio
n�
Rip
up
/Rero
ute
to
im
pro
ve s
olu
tio
n q
uality
(re
mo
ve a
seg
men
t an
d r
e-c
on
nect)
4
A
D
C
B
E
A
D
C
B
E
1
23
•R
esu
lts
are
sub
-op
tim
al
•In
her
it t
ime
and
mem
ory
co
st o
f m
aze
and
lin
e-p
rob
e al
go
rith
ms
Co
nn
ec
tin
g M
ult
i-T
erm
ina
l N
ets
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 9
1
Kah
ng
/Keu
tzer/
New
ton
Mu
lti-
term
inal
Nets
: D
iffe
ren
t R
ou
tin
g O
pti
on
s
(a)
Ste
iner
Tre
e (
14)
(b)
Ste
iner
Tre
e w
ith
Tru
nk
(15)
(c)
Min
imu
m S
pan
nin
gT
ree (
16)
(d)
Ch
ain
(17) (e)
Co
mp
lete
Gra
ph
(42)
Co
st is
det
erm
ined
by
rou
tin
g m
od
elC
ou
rtesy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 9
2
Kah
ng
/Keu
tzer/
New
ton
Ste
ine
r T
ree
Ba
se
d A
lgo
rith
ms
�T
ree inte
rconn
ectin
g a
set
of poin
ts (
dem
and p
oin
ts, D
) and s
om
e o
ther
(inte
rmedia
te)
poin
ts (
Ste
iner
poin
ts,
S)
�If S
is e
mpty
, S
tein
er
Min
imum
Tre
e (
SM
T)
equ
ivale
nt
to
Min
imum
Span
nin
g T
ree (
MS
T)
�F
indin
g S
MT
is N
P-c
om
ple
te;
many g
ood h
euristics
�S
MT
typic
ally
88%
of M
ST
cost; best heuristics a
re w
ithin
½%
of optim
al on a
vera
ge
�U
nderl
yin
g G
rid G
raph
defined b
y inte
rsectio
n o
f horizonta
l and v
ert
ica
l lin
es t
hro
ugh d
em
an
d p
oin
ts
(Han
an
gri
d) �
Rectilin
ear
SM
T a
nd M
ST
pro
ble
ms
�C
an m
odify M
ST
to a
ppro
xim
ate
RM
ST
, e.g
., b
uild
MS
T
and r
ectilin
earize
each e
dge
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 9
3
Kah
ng
/Keu
tzer/
New
ton
Min
imu
m S
pa
nn
ing
Tre
e (
Pri
m’s
co
ns
tru
cti
on
)
Giv
en
a w
eig
hte
d g
rap
h
Fin
d a
sp
an
nin
g t
ree w
ho
se w
eig
ht
is m
inim
um
Pri
m’s
alg
ori
thm
sta
rt w
ith
an
arb
itra
ry n
od
e s
T← ←←←
{s}
wh
ile T
is n
ot
a s
pan
nin
g t
ree
fin
d t
he c
losest
pair
x∈ ∈∈∈
V-T
, y
∈ ∈∈∈T
ad
d (
x,y
) to
T
run
s in
O(n
2)
tim
e
very
sim
ple
to
im
ple
men
t
alw
ays g
ives
a t
ree o
f m
inim
um
co
st
8
67
2
47
510
10
5
3
s45
2
89
γ γγγ
x
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 9
4
Kah
ng
/Keu
tzer/
New
ton
Ap
ply
ing
Sp
an
nin
g a
nd
Ste
ine
r T
ree
Alg
ori
thm
s
�G
en
era
l cell/b
lock d
esig
n:
ch
an
nel in
ters
ecti
on
gra
ph
s
�S
tan
dard
-cell
or
gate
-arr
ay
desig
n:
RS
MT
o
r R
MS
T in
geo
metr
y o
r g
rid
-gra
ph
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 9
5
Kah
ng
/Keu
tzer/
New
ton
Co
nc
urr
en
t G
lob
al
Ro
uti
ng
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 9
6
Kah
ng
/Keu
tzer/
New
ton
Glo
ba
l R
ou
tin
g:
Co
nc
urr
en
t A
pp
roa
ch
es
�C
an
fo
rmu
late
ro
utin
g p
rob
lem
as in
teg
er
pro
gra
mm
ing
, so
lve
sim
ulta
ne
ou
sly
fo
r a
ll n
ets
Giv
en
(i)
Set of S
tein
er
trees for
each n
et
(ii) P
lacem
ent of blo
cks/c
ells
(iii)
Channel capacitie
s
Dete
rmin
eS
ele
ct a S
tein
er
tree for
each n
et w
/o v
iola
ting c
hannel
capacitie
s
Optim
ize Min
tota
l w
irele
ngth
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 9
7
Kah
ng
/Keu
tzer/
New
ton
Yo
sh
imu
ra a
nd
Ku
h
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 9
8
Kah
ng
/Keu
tzer/
New
ton
Ho
rizo
nta
l C
on
str
ain
t G
rap
h (
HC
G)
1.
No
de v
i: r
ep
resen
ts a
ho
rizo
nta
l in
terv
al sp
an
ned
by n
et
i
2.
Th
ere
is a
n e
dg
e b
etw
een
vian
d v
jif
h
ori
zo
nta
l in
terv
als
overl
ap
3.
No
tw
o n
ets
wit
h a
ho
rizo
nta
l co
nstr
ain
t m
ay b
e a
ssig
ned
to
th
e s
am
e
track
4.
Maxim
um
cliq
ue o
f H
CG
esta
blish
es
low
er
bo
un
d o
n #
of
tracks:
# t
racks ≥ ≥≥≥
siz
e o
f m
axim
um
cliq
ue o
f H
CG
aa
a
c
a
bdef
ld(x
) e
a
dc
b
f
Lo
cal
den
sit
y a
t co
lum
n C
, ld
(C)
= #
nets
sp
lit
by c
olu
mn
C
Ch
an
nel D
en
sit
y d
= m
ax l
d(C
) o
ver
all
C
Each
net
sp
an
s o
ver
an
in
terv
al
Ho
rizo
nta
l C
on
str
ain
t G
rap
h(H
CG
) is
an
un
dir
ec
ted
gra
ph
wit
h:
ve
rtex :
net
ed
ge:
<n
_j,
n_k
>,
if i
nte
rva
ls I
_j,
I_k
in
ters
ec
tC
ou
rtesy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 9
9
Kah
ng
/Keu
tzer/
New
ton
Ve
rtic
al C
on
str
ain
t G
rap
h (
VC
G)
1.
No
de:
rep
resen
ts a
net
2.
Ed
ge (
a1
→ →→→a2)
exis
ts i
f at
so
me c
olu
mn
:
-N
et
a1 h
as a
term
inal
on
th
e u
pp
er
ed
ge
-N
et
a2 h
as a
term
inal
on
th
e l
ow
er
ed
ge
-E
dg
e a
1→ →→→
a2 m
ean
s t
hat
Net
a1
mu
st
be a
bo
ve N
et
a2
3.
Esta
blish
es lo
wer
bo
un
d:
# t
racks ≥ ≥≥≥
lon
gest
path
in
VC
G
4.
VC
G m
ay h
ave a
cycle
!
ab
bb
a
cc
a
cb
b
ba
a
ba
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
00
Kah
ng
/Keu
tzer/
New
ton
Do
gle
gs
in
Ch
an
ne
l R
ou
tin
g
Do
gle
gs m
ay r
ed
uce t
he lo
ng
est
path
in
VC
G
Do
gle
gs b
reak c
ycle
s in
VC
G
a b
c
d
d
a
b
c
a b c d
a b c-1
c-2
d
a
b
b
a
a
b
b
a
b-1
b-2
a b
b-1
b-2a
?
a b
c
d
d
a
b
c
c-1
c-2
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
01
Kah
ng
/Keu
tzer/
New
ton
Ch
ara
cte
rizin
g t
he
Ch
an
ne
l R
ou
tin
g P
rob
lem
1
35
89
267
10
41
3
54
8
9
10
7
6
2
Vert
ical
co
ns
tra
int
gra
ph
Gv
Ho
rizo
nta
l co
ns
train
t g
rap
h
Ch
an
nel
rou
tin
g p
rob
lem
is c
om
ple
tely
ch
ara
cte
rized
by t
he
ve
rtic
al
co
nstr
ain
t g
rap
h a
nd
th
e h
ori
zo
nta
l co
nstr
ain
t g
rap
h.
0
1
4
5
1
6
7
0
4
9
10
10
2
3
5
3
5
2
6
8
9
8
7
9
2
1
54
3
6
7
8
9
10
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
02
Kah
ng
/Keu
tzer/
New
ton
Th
eo
rem
A s
et
of
inte
rvals
wit
h d
en
sit
y d
can
be p
acked
in
to d
tra
ck
s.
Pro
of:
I 1
=(a
,b)
I
2=
(c,d
)D
efi
ne:
I1<
I 2if
fb
<c o
r I 1
=I 2
1.
refl
exiv
e:
I 1<
I 12.
an
ti-s
ym
metr
ic:
I 1<
I 2, I 2
<I 1
� ���I 1
=I 2
3.
tran
sit
ive
: I 1
<I 2
, I 2
<I 3� ���
I 1<
I 3
Set
of
inte
rvals
wit
h b
inary
rela
tio
n <
fo
rms a
part
iall
y o
rdere
d s
et
(PO
SE
T)
Inte
rvals
in
a s
ing
le t
rack� ���
form
a c
hain
Inte
rvals
in
ters
ecti
ng
a c
om
mo
n c
olu
mn
� ���fo
rm a
nan
tich
ain
Dilw
ort
h’s
th
eo
rem
(195
0):
If
the m
axim
um
an
tich
ain
of
a P
OS
ET
is o
f siz
e d
, th
en
th
e P
OS
ET
can
be p
art
itio
ned
in
to d
ch
ain
s
Inte
rva
l P
ac
kin
g
I 6
ca
bd
I 4I 3
I 1I 2
I 5
I 1
I 5I 2
I 3
I 4I 6
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
03
Kah
ng
/Keu
tzer/
New
ton
Le
ft-E
dg
e A
lgo
rith
m f
or
Inte
rva
l P
ac
kin
g
Rep
eat
cre
ate
a n
ew
tra
ck t
Rep
eat
pu
t le
ftm
ost
feasib
le in
terv
al to
t
un
tiln
o m
ore
feasib
le in
terv
al
un
tiln
o m
ore
in
terv
al
Inte
rvals
are
so
rted
acco
rdin
g t
o t
heir
left
en
dp
oin
ts
O(n
log
n)
tim
e a
lgo
rith
m.
Gre
ed
y a
lgo
rith
m w
ork
s!
I 6 I 4I 3
I 1I 2
I 5I 6 I 4
I 3
I 1I 2
I 5
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
Ho
rizo
nta
l C
on
str
ain
t G
rap
h (
HC
G)
�N
od
e v
i: r
ep
resen
ts a
ho
rizo
nta
l in
terv
al sp
an
ned
by n
et
I
�T
here
is a
n e
dg
e b
/w v
ian
d v
jif
h
ori
zo
nta
l in
terv
als
overl
ap
�N
o t
wo
nets
wit
h a
ho
rizo
nta
l co
nstr
ain
t m
ay b
e a
ssig
ned
to
th
e
sam
e t
rack
�M
axim
um
cliq
ue o
f H
CG
esta
blish
es a
lo
wer
bo
un
d o
n #
of
tracks:
# t
racks ≥ ≥≥≥
maxim
um
cliq
ue o
f H
CG
aa
a
c
a
bdef
ld(x
) e
a
dc
b
f
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
05
Kah
ng
/Keu
tzer/
New
ton
Ve
rtic
al C
on
str
ain
t G
rap
h (
VC
G)
�N
od
e:
rep
resen
ts a
net
�ed
ge (
a1
→ →→→a2):
if
at
so
me c
olu
mn
,
net
a1 h
as a
term
inal
on
th
e u
pp
er
ed
ge
net
a2 h
as a
term
inal
on
th
e l
ow
er
ed
ge
a1
→ →→→a2 m
ean
s t
hat
net
a1 h
as t
o
be a
bo
ve a
2
�E
sta
blish
es a
lo
wer
bo
un
d:
# t
racks ≥ ≥≥≥
lon
gest
path
in
VC
G
�V
CG
ma
y h
ave a
cycle
!
ab
bb
a
cc
a
cb
b
ba
a
ba
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
06
Kah
ng
/Keu
tzer/
New
ton
Do
gle
gs
in
Ch
an
ne
l R
ou
tin
g
�D
og
leg
s m
ay r
ed
uce t
he lo
ng
est
path
in
VC
G
�D
og
leg
s b
reak c
ycle
s in
VC
G
a b
c
d
d
a
b
c
a b c d
a b c-1
c-2
d
a
b
b
a
a
b
b
a
b-1
b-2
a b
b-1
b-2a
?
a b
c
d
d
a
b
c
c-1
c-2
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
07
Kah
ng
/Keu
tzer/
New
ton
Do
gle
gs
in
Ch
an
ne
l R
ou
tin
g(C
on
t’d
)
�R
estr
icte
d D
og
leg
vs
un
restr
icte
d d
og
leg
a
a
a
a
a
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
08
Kah
ng
/Keu
tzer/
New
ton
Co
ns
tra
int
Gra
ph
Ba
se
d A
lgo
rith
m:
“M
erg
ing
of
Ne
ts”
(Yo
sh
imu
ra &
Ku
h)
�O
n t
he a
ssum
ption o
f no c
yclic
constr
ain
ts,
nets
that
can b
e p
laced o
n
the s
am
e t
rack c
an b
e m
erg
ed in t
he V
CG
, sim
plif
yin
g t
he V
CG
.
�N
ets
can b
e o
rganiz
ed into
zo
nes,
furt
her
sim
plif
yin
g t
he p
roble
m
1
4
10
3
5 8
9
7
6
2
HIG
Zo
ne
12
34
5
1 2 3 4 56
7
9
10
8
Ch
ara
cte
rizin
g C
han
nel R
ou
tin
g P
rob
lem
1
35
89
267
10
41
3
54
8
9
10
7
6
2
Vert
ical
co
ns
tra
int
gra
ph
Gv
Ho
rizo
nta
l co
ns
train
t g
rap
h
Th
e c
han
nel
rou
tin
g p
rob
lem
is c
om
ple
tely
ch
ara
cte
rized
by t
he
ve
rtic
al
co
nstr
ain
t g
rap
h a
nd
th
e h
ori
zo
nta
l co
nstr
ain
t g
rap
h.
0
1
4
5
1
6
7
0
4
9
10
10
2
3
5
3
5
2
6
8
9
8
7
9
2
1
54
3
6
7
8
9
10
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
10
Kah
ng
/Keu
tzer/
New
ton
Zo
ne R
ep
resen
tati
on
of
Ho
rizo
nta
l S
eg
men
ts
0
1
4
5
1
6
7
0
4
9
10
1
0
2
3
5
3
5
2
6
8
9
8
7
9
2
1
54
3
67
8
9
10
2
1
1
1
1
2
4
4
4
7
7
92
2
2
2
4
6
7
7
8
9
10
3
3
3
4
6
7
8
8
9
10
4
4
5
9
5
5
Zo
ne
:
1
2
3
4
5
Zo
nes
are
max
imu
m c
liqu
es in
th
e h
ori
zon
tal i
nte
rval
gra
ph
.
Eac
h n
et m
ust
be
rou
ted
on
a d
iffe
ren
t tr
ack.
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
11
Kah
ng
/Keu
tzer/
New
ton
Me
rgin
g o
f N
ets
�D
efinitio
n:
Let
iand j
be n
ets
for
wh
ich t
he follo
win
g
ho
lds:
(a)
iand j
are
not adja
cent in
the H
IG
(b)
There
is n
o d
irect path
betw
een i
and j
in the V
CG
Then t
hese
nets
can b
e a
ssig
ne
d to the s
am
e t
rack
and h
ence t
hey c
an b
e m
erg
ed in t
he V
CG
�M
erg
ing O
pera
tio
n:
(1)
Com
bin
e n
odes i
and j
into
node i•
j in
VC
G
(2)
Update
zone r
epre
senta
tion s
uch that i•
joccupie
s z
ones o
f i
and j
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
12
Kah
ng
/Keu
tzer/
New
ton
Me
rgin
g o
f N
ets
: E
xa
mp
le
14
10
35
897 6 2
14
10
3
5• •••6
89
7
2
mer
ge
(5,6
)
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
13
Kah
ng
/Keu
tzer/
New
ton
Up
da
tin
g o
f Z
on
e R
ep
res
en
tati
on
Zo
ne
12
34
5
1 2 3 4 56
7
9
10
8
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
14
Kah
ng
/Keu
tzer/
New
ton
Me
rgin
g o
f N
ets
: E
xa
mp
le
14
10
35
897 6 2
14
10
3
5• •••6
89
7
2
mer
ge
(5,6
)
1• •••7
410
3
5• •••6
89
2
mer
ge
(1,7
)
1• •••7
4
10
3 • •••
8
5 • •••
6 • •••
9
2
mer
ge
(5,6
,9)
(3,8
)
1• •••7
10 • •••
4 3 • •••
8
5 • •••
6 • •••
9
2
(1)
(2) (3
) (4)
(5)
mer
ge
(4,1
0)
Fin
al R
ou
tin
g
0
1
4
5
1
6
7
0
4
9
10
10
2
3
5
3
5
2
6
8
9
8
7
9
2
154
3
6
7
8
9
10
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
16
Kah
ng
/Keu
tzer/
New
ton
Ro
uti
ng
Exam
ple
s b
y Y
-K’s
Alg
ori
thm
nu
mb
er
of
tra
cks
=18
max
imu
m d
en
sit
y =
18
nu
mb
er
of
tra
cks
=17
max
imu
m d
en
sit
y =
17
Exam
ple
3c
Exam
ple
4b
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
17
Kah
ng
/Keu
tzer/
New
ton
Ro
uti
ng
Exam
ple
s b
y Y
-K’s
Alg
ori
thm
(C
on
t’d
)
nu
mb
er
of
tra
cks
=20
max
imu
m d
en
sit
y =
20
nu
mb
er
of
tra
cks
=28
max
imu
m d
en
sit
y =
19
Exam
ple
5
Deu
tsch
’s D
iffi
cu
lt e
xa
mp
le w
ith
ou
t d
og
leg
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
18
Kah
ng
/Keu
tzer/
New
ton
Yo
sh
imu
ra a
nd
Ku
h’s
Me
tho
d
So
urc
e:
“E
ffic
ien
t A
lgo
rith
ms f
or
Ch
an
nel R
ou
tin
g”
by T
. Y
osh
imu
ra a
nd
E. K
uh
IEE
E T
ran
s.
On
Co
mp
ute
r-A
ided
Desig
n o
f In
teg
rate
d
Cir
cu
its a
nd
Syste
ms.
Vo
l. C
AD
-1, p
p25-3
5, Jan
1982
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
19
Kah
ng
/Keu
tzer/
New
ton
Co
mp
ac
tio
n
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
20
Kah
ng
/Keu
tzer/
New
ton
Co
mp
ac
tio
n
�C
han
nel C
om
pacti
on
( o
ne
dim
ensio
n)
�C
han
nel C
om
pacti
on
( o
ne
dim
ensio
n)
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
21
Kah
ng
/Keu
tzer/
New
ton
Co
mp
ac
tio
n
X-c
om
pactio
nX
-com
pactio
n
Y-c
om
pactio
nY
-com
pactio
n
�A
rea
Co
mp
actio
n(1
.5 o
r 2 d
imen
sio
n)
�A
rea
Co
mp
actio
n(1
.5 o
r 2 d
imen
sio
n)
�M
ay n
ee
d a
lo
t o
f co
nstr
ain
ts
to g
et
de
sir
ed
re
su
lts
�M
ay n
ee
d a
lo
t o
f co
nstr
ain
ts
to g
et
de
sir
ed
re
su
lts
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
22
Kah
ng
/Keu
tzer/
New
ton
Sh
ap
e-b
as
ed
Ro
uti
ng
�E
vo
lve
fro
m m
aze
ro
utin
g
�G
rid
less: lo
ok a
t a
ctu
al siz
e o
f e
ach
sh
ap
e
�E
ach
sh
ap
e m
ay h
ave
its
sp
acin
g r
ule
�G
oo
d fo
r d
esig
ns w
ith
mu
ltip
le w
idth
/sp
acin
g r
ule
s
an
d o
the
r com
ple
x r
ule
s
�S
low
er
tha
n g
rid
ded
ro
ute
r
So
urc
e
S2
S2
Targ
et
T2
T2
T1
T2
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
23
Kah
ng
/Keu
tzer/
New
ton
Oth
er
Ph
ys
ica
l Is
su
es
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
24
Kah
ng
/Keu
tzer/
New
ton
De
tail
ed
Ro
uti
ng
Ob
jec
tiv
es
�V
ia s
ele
ction
�V
ia a
rra
y b
ase
d o
n w
ire
siz
e o
r re
sis
tan
ce
�R
ecta
ng
ula
r via
ro
tatio
n a
nd
off
se
t
Rota
te a
nd o
ffset
horizonta
l via
s
No r
ota
tion f
or
a “
cro
ss”
via
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
25
Kah
ng
/Keu
tzer/
New
ton
De
tail
ed
Ro
uti
ng
Ob
jec
tiv
es
�U
nd
ers
tan
d c
om
ple
x p
in &
eq
uiv
ale
nt
pin
mo
de
ling
�U
nd
ers
tan
d c
om
ple
x p
in &
eq
uiv
ale
nt
pin
mo
de
ling
sim
ple
pin
Str
on
gS
tro
ng
Weak
Weak
Mu
st
Mu
st
com
ple
x p
in
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
26
Kah
ng
/Keu
tzer/
New
ton
Ex
tra
ex
tra
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
27
Kah
ng
/Keu
tzer/
New
ton
Ma
ze
Ro
uti
ng
�In
itia
lize p
riority
queu
e Q
, sourc
e S
, and s
ink T
�P
lace S
in Q
�G
et lo
west cost
poin
t X
fro
m Q
, put
neig
hbors
of X
in Q
�R
epe
at
last ste
p u
ntil lo
west-
cost po
int
X is e
qua
l to
the
sin
k T
�R
ip a
nd r
ero
ute
nets
, i.e., s
ele
ct
a n
um
ber
of nets
based
on a
cost fu
nctio
n (
e.g
., c
ongestion o
f re
gio
ns t
hro
ugh
wh
ich n
et tr
ave
ls),
then r
em
ove the n
et
and r
ero
ute
it
�M
ain
obje
ctive: r
educe o
verf
low
�E
dge o
verf
low
= 0
if num
_nets
less than o
r equal to
the c
apacity
�E
dge o
verf
low
= n
um
_nets
–capacity if num
_nets
is g
reate
r th
an c
apacity
�O
verf
low
= Σ
(edge o
verf
low
s)
over
all
edges
Co
urt
esy K
. K
eu
tze
r e
t al. U
CB
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
28
Kah
ng
/Keu
tzer/
New
ton
Mik
am
i&
Ta
bu
ch
i’s
Alg
ori
thm
D.
Pan
–U
niv
ers
ity o
f T
exa
s
EC
E 2
60B
–C
SE
241
A /U
CB
EE
CS
244 1
29
Kah
ng
/Keu
tzer/
New
ton
Ma
ze
Ro
uti
ng
Co
st
Fu
nc
tio
n a
nd
Dir
ec
ted
Se
arc
h
�P
oin
ts c
an b
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