parallelizing the simplex method with...
TRANSCRIPT
01
23
45
67
x 10
7
0
102030405060708090
100
Pro
ble
m S
ize
Memory Bandwidth Usage (%)
FP
GA
GP
UC
PU
Op
enC
LC
PU
Op
enC
LC
PU
Op
enC
LC
PU
Op
enC
LC
PU
Op
enC
L
105
106
107
108
100
101
102
Pro
ble
m S
ize
Speed Up
FP
GA
vs
CP
U
GP
U v
s C
PU
CP
U O
pen
CL
vs
CP
U
GP
U v
s So
Ple
x
10
51
06
10
71
08
10
3
10
4
10
5
Pro
ble
m S
ize
Iterations per Unit Energy (mJ-1
)
FP
GA
GP
U
CP
U O
pen
CL
CP
U
App
lic
at
ion
: Ra
dia
tio
n T
her
ap
y T
rea
tm
ent
Pla
nn
ing
• R
adia
tio
n t
her
apy
use
s se
vera
l bea
ms
to d
eliv
er a
rad
iati
on
do
se t
o t
um
ou
r ce
lls.
Rec
ent
lite
ratu
re h
as f
orm
ula
ted
th
e p
rob
lem
of
op
tim
izin
g th
e b
eam
wei
ghts
as
an
L
P p
rob
lem
[3-
4].
• �
e a�
ecte
d a
rea
is d
ivid
ed i
nto
tar
get t
and
no
rmal
n v
oxe
ls.
�e
rad
iati
on
fro
m e
ach
bea
m t
o e
ach
vo
xel i
s m
easu
red
to
fo
rm t
he
do
se m
atri
x D
.
• �
e m
od
el (
2) m
inim
izes
th
e to
tal d
ose
del
iver
ed t
o n
on
-tar
get
area
s su
bje
ct t
o
co
nst
rain
ts b
ased
on
th
e p
rop
erti
es o
f th
e ti
ssu
e in
eac
h v
oxe
l.
• V
ecto
rs x
rep
rese
nt
the
up
per
U a
nd
low
er L
b
ou
nd
s fo
r th
e ta
rget
an
d n
orm
al a
reas
. Bea
m
wei
ghts
w a
re g
iven
co
sts cN.
• Si
nce
eac
h b
eam
wil
l del
iver
rad
iati
on
to
mo
st
voxe
ls, t
he
do
se m
atri
x o
f th
e L
P p
rob
lem
w
ill b
e d
ense
.
min
(DT ncN)T
w
Dtw
≤xU t
Dtw
≥xL t
Dnw
≤xU n
w≥
0
solu
tio
ns
(BF
S),
of
the
convex
po
lyh
edro
nre
pre
sen
tin
gth
e
Per
for
ma
nc
e B
enc
hm
ar
kin
g
• L
angu
age
des
ign
ed b
y K
hro
no
s G
rou
p f
or
po
rtab
ilit
y ac
ross
har
dw
are
acce
lera
tors
.
• A
Har
dw
are
acce
lera
tor,
or
Op
enC
L D
evic
e, i
s m
anag
ed b
y a
Ho
st p
roce
sso
r.
• �
e O
pen
CL
lin
ear
pro
gram
min
g en
gin
e w
as t
este
d o
n m
ult
iple
dev
ices
to
mea
sure
acce
lera
tio
n (
see
Tab
le 1
fo
r sp
eci#
cati
on
s).
• E
�o
rt w
as m
ade
to o
pti
miz
e O
pen
CL
ker
nel
s fo
r ea
ch d
evic
e w
ith
ou
t lo
sin
g d
esig
n
p
ort
abil
ity.
• A
het
ero
gen
ou
s O
pen
CL
sys
tem
is
pro
po
sed
to
so
lve
den
se L
P p
rob
lem
s an
d
te
sted
on
mu
ltip
le h
ard
war
e ac
cele
rato
rs.
• T
est
resu
lts
ind
icat
e th
at t
he
GP
U p
rovi
des
th
e fa
stes
t so
luti
on
s to
LP
pro
ble
ms
as
sh
ow
n i
n F
igu
re 3
.
• �
e se
lect
ed F
PG
A p
rovi
des
su
per
ior
ener
gy
e$ci
ency
fo
r th
e ap
pli
cati
on
as
sh
ow
n i
n F
igu
re 4
.
• �
e Si
mp
lex
Alg
ori
thm
so
lves
LP
pro
ble
ms
by
iter
atin
g o
ver
sub
sets
of
the
pro
ble
m
va
riab
les
to g
ener
ate
imp
rovi
ng
solu
tio
ns.
• E
ach
sim
ple
x it
erat
ion
co
nsi
sts
of
thre
e su
bro
uti
nes
: pri
cin
g, r
atio
tes
t, a
nd
piv
ot.
• P
ivo
t #
nd
s th
e d
irec
tio
n o
f th
e ad
jace
nt
solu
tio
n, r
atio
tes
t ca
lcu
late
s th
e m
agn
itu
de
of
th
e re
du
ctio
n i
n t
he
cost
fu
nct
ion
, an
d p
ivo
t u
pd
ates
th
e p
rob
lem
dat
a st
ruct
ure
s.
• �
e th
ree
sub
rou
tin
es a
re p
erfo
rmed
on
th
e O
pen
CL
Dev
ice
in a
loo
p m
anag
ed b
y th
e
Ho
st p
roce
sso
r.
Device
De
De
Power
(W)
Mem
ory
Ban
dwidth
Ban
dBan
d(G
B/s)
Inte
lC
ore
i749
30k
(6co
res)
130
59.7
Nvi
dia
GeF
orc
eG
TX
-780
250
288.
4
Alt
era
Nal
late
chP
CIe
-385
N25
24.
0
Int
ro
du
ct
ion
HA
RD
WA
RE A
CC
EL
ER
AT
ED
LIN
EA
R P
RO
GR
AM
MIN
GP
aral
leli
zin
g th
e Si
mp
lex
Met
ho
d w
ith
Op
enC
L
Th
e Si
mpl
ex A
lgo
rit
hm
• L
inea
r P
rogr
amm
ing
(LP
) m
od
els
are
freq
uen
tly
enco
un
tere
d o
pti
miz
atio
n
pro
ble
ms
in m
edic
ine,
en
gin
eeri
ng
and
bu
sin
ess
, wh
ere
a li
nea
r o
bje
ctiv
e fu
nct
ion
nee
ds
to b
e ei
ther
min
imiz
ed o
r m
axim
ized
su
bje
ct t
o m
ult
iple
lin
ear
con
stra
ints
.
• H
igh
per
form
ance
im
ple
men
tati
on
s o
f th
e Si
mp
lex
Alg
ori
thm
fac
ilit
ate
solu
tio
ns
to
larg
e p
rob
lem
s in
th
ese
dis
cip
lin
es.
• �
is w
ork
pro
po
ses
an e
ner
gy-
e$ci
ent
har
dw
are
acce
lera
ted
LP
so
lver
fo
r a
clas
s
of
pro
ble
ms
in r
adia
tio
n t
her
apy.
�e
pro
po
sed
acc
eler
ato
r is
im
ple
men
ted
an
d
te
sted
on
var
iou
s d
evic
es u
sin
g O
pen
CL
.
[1]
A. H
amzi
c, A
. Hu
sein
ovi
c, a
nd
N. N
oso
vic,
“Im
ple
men
tati
on
an
d p
erfo
rman
ce a
nal
ysis
of
the
sim
ple
x al
gori
thm
ad
apte
d t
o r
un
on
co
mm
od
ity
op
encl
en
able
d g
rap
hic
s p
roce
sso
rs,”
in
201
1 X
XII
I In
tern
atio
nal
Sym
pos
ium
on
In
form
atio
n, C
omm
un
icat
ion
an
d
A
uto
mat
ion
Tec
hn
olog
ies
(IC
AT
), 2
011,
pp
. 1–
7.
[2]
S. B
ayli
ss, C
.-S.
Bo
uga
nis
, G. C
on
stan
tin
ides
, an
d W
. Lu
k, “
An
fp
ga i
mp
lem
enta
tio
n o
f th
e si
mp
lex
algo
rith
m,”
in
IE
EE
In
tern
atio
nal
C
onfe
ren
ce o
n F
ield
Pro
gram
mab
le T
ech
nol
ogy,
200
6, p
p. 4
9–56
.
[3]
A. O
lafs
son
an
d S
. Wri
ght,
“L
inea
r p
rogr
amm
ing
form
ula
tio
ns
and
alg
ori
thm
s fo
r ra
dio
ther
apy
trea
tmen
t p
lan
nin
g,”
Op
tim
izat
ion
M
eth
ods
and
Sof
twar
e, v
ol.
21, n
o. 2
, pp
. 201
–23
1, A
pri
l 200
6.
[4]
H
. Ro
mei
jn, R
. Ah
uja
, J. D
emp
sey,
an
d A
. Ku
mar
, “A
new
lin
ear
pro
gram
min
g ap
pro
ach
to
rad
iati
on
th
erap
y tr
eatm
ent
pla
nn
ing
p
rob
lem
s,” O
per
atio
ns
Res
earc
h, v
ol.
54, n
o. 2
, pp
. 201
–21
6, A
pri
l 200
6.
[5]
R. W
un
der
lin
g, “
Par
alle
ler
un
d o
bje
kto
rien
tier
ter
Sim
ple
x-A
lgo
rith
mu
s,” P
h.D
. dis
sert
atio
n, T
ech
nis
che
Un
iver
sita
t B
erli
n,
A
lgo
rith
mu
s,” P
h.D
. dis
sert
atio
n, T
ech
nis
che
Un
iver
sit¨
at B
erli
n, 1
996,
htt
p:/
/ww
w.z
ib.d
e/P
ub
lica
tio
ns/
abst
ract
s/T
R-9
6-09
/.
Co
nc
lusi
on
s
Bra
dle
y d
e V
lugt
, May
sam
Mir
ahm
adi,
Ser
guei
L. P
rim
ak, A
bd
alla
h S
ham
i
Fig
ure
3: S
pee
d U
p f
or
the
Op
enC
L S
olv
er O
ver
the
Seq
uen
tial
So
lver
Fig
ure
4: E
ner
gy
E$
cien
cy o
f th
e O
pen
CL
and
Seq
uen
tial
So
lver
s
Ope
nC
LD
esig
n A
na
lysi
s
Tab
le 1
: Op
enC
L T
est
Dev
ice
Spec
i#ca
tio
ns
• P
erfo
rman
ce b
ench
mar
kin
g re
veal
s sp
eed
up
s re
lati
ve t
o s
equ
enti
al c
od
e th
at
app
roac
h 2
an
d 1
0 o
n a
CP
U a
nd
GP
U.
• �
e F
PG
A e
xhib
its
clo
se t
o u
nit
y sp
eed
up
bu
t p
rove
d t
o b
e th
e m
ost
e$
cien
t in
term
s o
f Si
mp
lex
iter
atio
ns
pro
cess
ed p
er u
nit
en
erg
y w
ith
an
e$
cien
cy 5
tim
es
gr
eate
r th
an t
he
CP
U.
• �
e d
esig
n e
�ec
tive
ly s
atu
rate
s th
e m
emo
ry b
and
wid
th o
f ea
ch t
est
dev
ice
as s
ho
wn
in F
igu
re 5
.
min
cx
s.t.
Ax≤
b
x≥
0
Lin
ear
Pr
og
ra
mm
ing
• �
e st
and
ard
fo
rm o
f a
lin
ear
pro
gram
min
g p
rob
lem
is
(1),
wh
ere A
rep
rese
nts
th
e
con
stra
int
syst
em, b
rep
rese
nts
th
e co
nst
rain
t b
ou
nd
s, c
is
the
cost
fu
nct
ion
, an
d x
is t
he
dec
isio
n v
aria
ble
s.
(1)
(2)
Star
tSi
mp
lex
Tra
nsf
erP
rob
lem
Pri
cin
g
IsB
asis
Op
tim
al?
Rat
ioT
est
TT
Piv
oP
iP
it
Rea
dSo
luti
on
Sto
pSi
mp
lex
no
yes
Ho
st
Dev
ice
De
De
Fig
ure
2: O
pen
CL
Sim
ple
x A
lgo
rith
m D
esig
n
Op
tim
al S
olu
tio
n
Bas
ic F
easi
ble
So
luti
on
Po
ssib
le S
imp
lex
Pat
h
Fea
sib
le R
egio
n
Co
nst
rain
t L
ine
a) �
e Si
mp
lex
Alg
ori
thm
b)
Op
enC
L I
mp
lem
enta
tio
n
Fig
ure
5: M
emo
ry B
and
wid
th U
sage
Fig
ure
1: R
adia
tio
n �
erap
yB
eam
Co
n#
gura
tio
n [
4]
Ref
eren
ces
Ac
kn
ow
led
gem
ent
�is
wo
rk w
as s
up
po
rted
in
par
t b
y th
e S
ou
ther
n O
nta
rio
Sm
art
Co
mp
uti
ng
Inn
ova
tio
n P
latf
orm
Co
nso
rtiu
m (
SOSC
IP)
and
IB
M C
anad
a L
td.
�e
auth
ors
wo
uld
lik
e to
th
ank
Dr.
Mar
y F
enel
on
, Dr.
Sea
n W
agn
er, M
r.
Las
zlo
Lad
anyi
an
d M
r. B
lair
Ad
amac
he
fro
m I
BM
fo
r th
eir
valu
able
co
mm
ents
an
d s
ugg
esti
on
s.