csr2011 june14 11_30_winzen
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TRANSCRIPT
To
wa
rds
a C
om
ple
xit
y T
he
ory
fo
r
Ra
nd
om
ize
d S
ea
rch
He
uri
sti
cs
: T
he
Ra
nk
ing
-Ba
se
d B
lac
k-B
ox
Mo
de
l
Be
nja
min
Do
err
/ C
aro
la W
inze
n
CS
R,
Ju
ne
14
, 2
01
1
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ou
r M
oti
va
tio
n
G
en
era
l-pu
rpose
(ran
do
miz
ed
) searc
h h
eu
risti
cs a
re
...e
asy
to im
ple
ment,
...v
ery
fle
xib
le,
...n
ee
d little a
priori k
now
led
ge a
bo
ut
pro
ble
m a
t ha
nd
,
...c
an d
eal w
ith m
any c
onstr
ain
ts a
nd n
onlin
ea
rities,
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ou
r M
oti
va
tio
n
G
en
era
l-pu
rpose
(ran
do
miz
ed
) searc
h h
eu
risti
cs a
re
...e
asy
to im
ple
ment,
...v
ery
fle
xib
le,
...n
ee
d little a
priori k
now
led
ge a
bo
ut
pro
ble
m a
t ha
nd
,
...c
an d
eal w
ith m
any c
onstr
ain
ts a
nd n
onlin
ea
rities,
and
th
us, fr
equ
en
tly a
pplie
d in p
ractice.
S
om
e t
he
ore
tical re
su
lts e
xis
t.
T
ypic
al re
sult:
runtim
e a
naly
sis
fo
r
a p
art
icula
r alg
orith
m
on a
part
icula
r p
roble
m.
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ou
r M
oti
va
tio
n
G
en
era
l-pu
rpose
(ran
do
miz
ed
) searc
h h
eu
risti
cs a
re
...e
asy
to im
ple
ment,
...v
ery
fle
xib
le,
...n
ee
d little a
priori k
now
led
ge a
bo
ut
pro
ble
m a
t ha
nd
,
...c
an d
eal w
ith m
any c
onstr
ain
ts a
nd n
onlin
ea
rities,
and
th
us, fr
equ
en
tly a
pplie
d in p
ractice.
S
om
e t
he
ore
tical re
su
lts e
xis
t.
T
ypic
al re
sult:
runtim
e a
naly
sis
fo
r
a p
art
icula
r alg
orith
m
on a
part
icula
r p
roble
m.
Com
pa
rable
to
th
e “
ea
rly d
ays”
of C
om
pute
r S
cie
nce
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ou
r M
oti
va
tio
n
In
cla
ssic
al th
eo
retical com
pu
ter
scie
nce:
firs
t re
sults: ru
ntim
e a
naly
sis
fo
r
a p
art
icula
r alg
orith
m
on a
part
icula
r p
roble
m.
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ou
r M
oti
va
tio
n
In
cla
ssic
al th
eo
retical com
pu
ter
scie
nce:
firs
t re
sults: ru
ntim
e a
naly
sis
fo
r
a p
art
icula
r alg
orith
m
on a
part
icula
r p
roble
m.
gen
era
l lo
wer
bo
und
s (
“tra
cta
bili
ty o
f a p
roble
m”)
com
ple
xity t
he
ory
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ou
r M
oti
va
tio
n
In
cla
ssic
al th
eo
retical com
pu
ter
scie
nce:
firs
t re
sults: ru
ntim
e a
naly
sis
fo
r
a p
art
icula
r alg
orith
m
on a
part
icula
r p
roble
m.
gen
era
l lo
wer
bo
und
s (
“tra
cta
bili
ty o
f a p
roble
m”)
com
ple
xity t
he
ory
How
to c
reate
a
com
ple
xity t
he
ory
fo
r ra
nd
om
ize
d s
earc
h
heu
ristics?
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ou
r M
oti
va
tio
n
In
cla
ssic
al th
eo
retical com
pu
ter
scie
nce:
firs
t re
sults: ru
ntim
e a
naly
sis
fo
r
a p
art
icula
r alg
orith
m
on a
part
icula
r p
roble
m.
gen
era
l lo
wer
bo
und
s (
“tra
cta
bili
ty o
f a p
roble
m”)
com
ple
xity t
he
ory
O
ur
aim
:to
und
ers
tand t
he
tra
cta
bil
ity o
f a p
rob
lem
for
ge
nera
l-p
urp
ose
(ra
nd
om
ize
d)
searc
h h
eu
ris
tic
s
“Tow
ard
s a
Com
ple
xity T
he
ory
fo
r R
and
om
ize
d S
ea
rch H
eu
ristics”
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
A G
en
era
l V
iew
on
Se
arc
h H
eu
ris
tic
s
Searc
h
Heuristic
Bla
ck-B
ox
= “
Ora
cle
”
f fff
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
A G
en
era
l V
iew
on
Se
arc
h H
eu
ris
tic
s
Searc
h
Heuristic
Bla
ck-B
ox
= “
Ora
cle
”
x xxx
f fff
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
A G
en
era
l V
iew
on
Se
arc
h H
eu
ris
tic
s
Searc
h
Heuristic
Bla
ck-B
ox
= “
Ora
cle
”
x xxx
f fff(x xxx
)
f fff
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
A G
en
era
l V
iew
on
Se
arc
h H
eu
ris
tic
s
Searc
h
Heuristic
Bla
ck-B
ox
= “
Ora
cle
”
x xxx
f fff(x xxx
)
f fff
Typic
al cost m
easure
: n
um
ber
of
fun
cti
on
eva
lua
tio
ns
until a
n o
pti
mal s
olu
tio
nis
qu
eri
ed f
or
the fir
st tim
e
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
A G
en
era
l V
iew
on
Se
arc
h H
eu
ris
tic
s
Searc
h
Heuristic
Bla
ck-B
ox
= “
Ora
cle
”
x xxx
f fff(x xxx
)
f fff
Typic
al cost m
easure
: n
um
ber
of
fun
cti
on
eva
lua
tio
ns
until a
n o
pti
mal s
olu
tio
nis
qu
eri
ed f
or
the fir
st tim
e
Cla
ss
ical
Qu
ery
Co
mp
lexit
y M
od
el
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
A T
he
ory
fo
r (R
an
do
miz
ed
) S
ea
rch
He
uri
sti
cs
P
art
1:C
lassic
al
qu
ery
co
mp
lexit
y m
od
el
G
am
e t
he
ore
tic v
iew
E
xam
ple
: M
aste
rmin
d
P
art
2: R
efine
me
nt: r
an
kin
g-b
as
ed
qu
ery
co
mp
lexit
y
“Tow
ard
s a
Com
ple
xity T
he
ory
fo
r R
and
om
ize
d S
ea
rch H
eu
ristics:
Th
e R
ankin
g-B
ased B
lack-B
ox M
odel”
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ex
am
ple
: A
Ma
ste
rmin
d P
rob
lem
C
aro
le(=
ora
cle
) ch
oo
ses a
bin
ary
str
ing o
f le
ng
th n
:
11
01
00
11
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ex
am
ple
: A
Ma
ste
rmin
d P
rob
lem
C
aro
le(=
ora
cle
) ch
oo
ses a
bin
ary
str
ing o
f le
ng
th n
:
P
aul(=
alg
o.)
tries t
o f
ind it. H
e m
ay a
sk a
ny s
trin
g o
f le
ngth
n:
11
01
00
11
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ex
am
ple
: A
Ma
ste
rmin
d P
rob
lem
C
aro
le(=
ora
cle
) ch
oo
ses a
bin
ary
str
ing o
f le
ng
th n
:
P
aul(=
alg
o.)
tries t
o f
ind it. H
e m
ay a
sk a
ny s
trin
g o
f le
ngth
n:
11
01
00
11
10
10
10
10
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ex
am
ple
: A
Ma
ste
rmin
d P
rob
lem
C
aro
le(=
ora
cle
) ch
oo
ses a
bin
ary
str
ing o
f le
ng
th n
:
P
aul(=
alg
o.)
tries t
o f
ind it. H
e m
ay a
sk a
ny s
trin
g o
f le
ngth
n:
C
aro
leco
mp
ute
s in h
ow
ma
ny p
ositio
ns th
e s
trin
gs c
oin
cid
e:
11
01
00
11
10
10
10
10
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ex
am
ple
: A
Ma
ste
rmin
d P
rob
lem
C
aro
le(=
ora
cle
) ch
oo
ses a
bin
ary
str
ing o
f le
ng
th n
:
P
aul(=
alg
o.)
tries t
o f
ind it. H
e m
ay a
sk a
ny s
trin
g o
f le
ngth
n:
C
aro
leco
mp
ute
s in h
ow
ma
ny p
ositio
ns th
e s
trin
gs c
oin
cid
e:
11
01
00
11
10
10
10
10
10
10
10
10
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ex
am
ple
: A
Ma
ste
rmin
d P
rob
lem
C
aro
le(=
ora
cle
) ch
oo
ses a
bin
ary
str
ing o
f le
ng
th n
:
P
aul(=
alg
o.)
tries t
o f
ind it. H
e m
ay a
sk a
ny s
trin
g o
f le
ngth
n:
C
aro
leco
mp
ute
s in h
ow
ma
ny p
ositio
ns th
e s
trin
gs c
oin
cid
e:
“P
aul, o
ur
str
ings c
oin
cid
e in 3
bits”
11
01
00
11
10
10
10
10
10
10
10
10
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ex
am
ple
: A
Ma
ste
rmin
d P
rob
lem
C
aro
le(=
ora
cle
) ch
oo
ses a
bin
ary
str
ing o
f le
ng
th n
:
P
aul(=
alg
o.)
tries t
o f
ind it. H
e m
ay a
sk a
ny s
trin
g o
f le
ngth
n:
C
aro
leco
mp
ute
s in h
ow
ma
ny p
ositio
ns th
e s
trin
gs c
oin
cid
e:
“P
aul, o
ur
str
ings c
oin
cid
e in 3
bits”
11
01
00
11
10
10
10
10
10
10
10
10
We s
ay t
ha
t th
e
“fi
tnes
s”
of
Paul’s
str
ing is
3
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ex
am
ple
: A
Ma
ste
rmin
d P
rob
lem
C
aro
lech
ooses a
bin
ary
str
ing o
f le
ngth
n:
P
aultr
ies t
o fin
d it. H
e m
ay a
sk a
ny s
trin
g o
f le
ng
th n
:
11
01
00
11
10
10
10
10
3
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ex
am
ple
: A
Ma
ste
rmin
d P
rob
lem
C
aro
lech
ooses a
bin
ary
str
ing o
f le
ngth
n:
P
aultr
ies t
o fin
d it. H
e m
ay a
sk a
ny s
trin
g o
f le
ng
th n
:
11
01
00
11
10
10
10
10
3
10
11
01
01
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ex
am
ple
: A
Ma
ste
rmin
d P
rob
lem
C
aro
lech
ooses a
bin
ary
str
ing o
f le
ngth
n:
P
aultr
ies t
o fin
d it. H
e m
ay a
sk a
ny s
trin
g o
f le
ng
th n
:
11
01
00
11
10
10
10
10
3
10
11
01
01
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ex
am
ple
: A
Ma
ste
rmin
d P
rob
lem
C
aro
lech
ooses a
bin
ary
str
ing o
f le
ngth
n:
P
aultr
ies t
o fin
d it. H
e m
ay a
sk a
ny s
trin
g o
f le
ng
th n
:
11
01
00
11
10
10
10
10
3 4
...
10
11
01
01
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ex
am
ple
: A
Ma
ste
rmin
d P
rob
lem
C
aro
lech
ooses a
bin
ary
str
ing o
f le
ngth
n:
P
aultr
ies t
o fin
d it. H
e m
ay a
sk a
ny s
trin
g o
f le
ng
th n
:
11
01
00
11
10
10
10
10
3 4
...
How
ma
ny q
ue
ries d
oes P
auln
eed,
on
ave
rage
, until h
e h
as iden
tified C
aro
le’s
str
ing?
10
11
01
01
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Re
min
de
r
O
ur
aim
: T
o u
nd
ers
tand t
rac
tab
ilit
y o
f a p
rob
lem
for
gene
ral-p
urp
ose (
rand
om
ized
) se
arc
h h
euristics
M
easu
re:
nu
mb
er
of
fun
cti
on
eva
lua
tio
ns
until
an o
ptim
al solu
tio
n is q
ue
rie
d fo
r th
e fir
st tim
e
O
ur
main
inte
rest: g
oo
d l
ow
er
bo
un
ds
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e M
as
ter
Min
d P
rob
lem
: W
hat
Se
arc
h H
eu
risti
cs D
o
P
aultr
ies t
o fin
d C
aro
le’s
bin
ary
str
ing o
f le
ngth
n:
11
01
00
11
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e M
as
ter
Min
d P
rob
lem
: W
hat
Se
arc
h H
eu
risti
cs D
o
P
aultr
ies t
o fin
d C
aro
le’s
bin
ary
str
ing o
f le
ngth
n:
F
irst q
ue
ry is a
rbitra
ry:
11
01
00
11
10
10
10
10
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e M
as
ter
Min
d P
rob
lem
: W
hat
Se
arc
h H
eu
risti
cs D
o
P
aultr
ies t
o fin
d C
aro
le’s
bin
ary
str
ing o
f le
ngth
n:
F
irst q
ue
ry is a
rbitra
ry:
11
01
00
11
10
10
10
10
3
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e M
as
ter
Min
d P
rob
lem
: W
hat
Se
arc
h H
eu
risti
cs D
o
P
aultr
ies t
o fin
d C
aro
le’s
bin
ary
str
ing o
f le
ngth
n:
F
irst q
ue
ry is a
rbitra
ry:
T
hen
flip
exactly o
ne b
it (
chosen
u.a
.r.)
:
11
01
00
11
10
10
10
10
3
11
10
10
10
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e M
as
ter
Min
d P
rob
lem
: W
hat
Se
arc
h H
eu
risti
cs D
o
P
aultr
ies t
o fin
d C
aro
le’s
bin
ary
str
ing o
f le
ngth
n:
F
irst q
ue
ry is a
rbitra
ry:
T
hen
flip
exactly o
ne b
it (
chosen
u.a
.r.)
:
11
01
00
11
10
10
10
10
3
11
10
10
10
4
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e M
as
ter
Min
d P
rob
lem
: W
hat
Se
arc
h H
eu
risti
cs D
o
P
aultr
ies t
o fin
d C
aro
le’s
bin
ary
str
ing o
f le
ngth
n:
F
irst q
ue
ry is a
rbitra
ry:
T
hen
flip
exactly o
ne b
it (
chosen
u.a
.r.)
:
A
nd it co
ntinues w
ith t
he b
ett
er
of th
e tw
o:
11
01
00
11
10
10
10
10
3
11
10
10
10
4
11
10
11
10
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e M
as
ter
Min
d P
rob
lem
: W
hat
Se
arc
h H
eu
risti
cs D
o
P
aultr
ies t
o fin
d C
aro
le’s
bin
ary
str
ing o
f le
ngth
n:
F
irst q
ue
ry is a
rbitra
ry:
T
hen
flip
exactly o
ne b
it (
chosen
u.a
.r.)
:
11
01
00
11
10
10
10
10
3
11
10
10
10
4
Ra
nd
om
Lo
ca
l S
ea
rch
alg
ori
thm
:
Θ(nlogn)
Coup
on C
olle
cto
r [F
olk
lore
]
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e M
as
ter
Min
d P
rob
lem
: W
hat
Se
arc
h H
eu
risti
cs D
o
P
aultr
ies t
o fin
d C
aro
le’s
bin
ary
str
ing o
f le
ngth
n:
F
irst q
ue
ry is a
rbitra
ry:
11
01
00
11
10
10
10
10
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e M
as
ter
Min
d P
rob
lem
: W
hat
Se
arc
h H
eu
risti
cs D
o
P
aultr
ies t
o fin
d C
aro
le’s
bin
ary
str
ing o
f le
ngth
n:
F
irst q
ue
ry is a
rbitra
ry:
F
lip e
ach b
it w
ith p
rob
abili
ty 1
/n:
11
01
00
11
10
10
10
10
3
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e M
as
ter
Min
d P
rob
lem
: W
hat
Se
arc
h H
eu
risti
cs D
o
P
aultr
ies t
o fin
d C
aro
le’s
bin
ary
str
ing o
f le
ngth
n:
F
irst q
ue
ry is a
rbitra
ry:
F
lip e
ach b
it w
ith p
rob
abili
ty 1
/n:
11
01
00
11
10
10
10
10
3
00
10
11
10
1
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e M
as
ter
Min
d P
rob
lem
: W
hat
Se
arc
h H
eu
risti
cs D
o
P
aultr
ies t
o fin
d C
aro
le’s
bin
ary
str
ing o
f le
ngth
n:
F
irst q
ue
ry is a
rbitra
ry:
F
lip e
ach b
it w
ith p
rob
abili
ty 1
/n:
A
nd it co
ntinues w
ith t
he b
ett
er
of th
e tw
o
11
01
00
11
10
10
10
10
3
00
10
11
10
1
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e M
as
ter
Min
d P
rob
lem
: W
hat
Se
arc
h H
eu
risti
cs D
o
P
aultr
ies t
o fin
d C
aro
le’s
bin
ary
str
ing o
f le
ngth
n:
F
irst q
ue
ry is a
rbitra
ry:
F
lip e
ach b
it w
ith p
rob
abili
ty 1
/n:
11
01
00
11
10
10
10
10
3
00
10
11
10
1
(1+
1)
Evo
lutio
na
ry A
lgo
rith
m:
Θ(nlogn)
[Mü
hle
nbe
in9
2]
[Dro
ste
/Jan
sen/W
ege
ner
02
]
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e M
as
ter
Min
d P
rob
lem
: O
pti
mal S
tra
teg
ies (
1/2
)
P
aultr
ies t
o fin
d C
aro
le’s
bin
ary
str
ing o
f le
ngth
n:
10
01
00
11
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e M
as
ter
Min
d P
rob
lem
: O
pti
mal S
tra
teg
ies (
1/2
)
P
aultr
ies t
o fin
d C
aro
le’s
bin
ary
str
ing o
f le
ngth
n:
H
e c
an
go
thro
ugh t
he s
trin
g b
it b
y b
it:
10
01
00
11
00
00
00
00
4
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e M
as
ter
Min
d P
rob
lem
: O
pti
mal S
tra
teg
ies (
1/2
)
P
aultr
ies t
o fin
d C
aro
le’s
bin
ary
str
ing o
f le
ngth
n:
H
e c
an
go
thro
ugh t
he s
trin
g b
it b
y b
it:
10
01
00
11
00
00
00
00
4
10
00
00
00
5
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e M
as
ter
Min
d P
rob
lem
: O
pti
mal S
tra
teg
ies (
1/2
)
P
aultr
ies t
o fin
d C
aro
le’s
bin
ary
str
ing o
f le
ngth
n:
H
e c
an
go
thro
ugh t
he s
trin
g b
it b
y b
it:
10
01
00
11
00
00
00
00
4
10
00
00
00
5
11
00
00
00
4
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e M
as
ter
Min
d P
rob
lem
: O
pti
mal S
tra
teg
ies (
1/2
)
P
aultr
ies t
o fin
d C
aro
le’s
bin
ary
str
ing o
f le
ngth
n:
H
e c
an
go
thro
ugh t
he s
trin
g b
it b
y b
it:
10
01
00
11
00
00
00
00
4
10
00
00
00
5
11
00
00
00
4
...
10
01
00
11
8
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e M
as
ter
Min
d P
rob
lem
: O
pti
mal S
tra
teg
ies (
1/2
)
P
aultr
ies t
o fin
d C
aro
le’s
bin
ary
str
ing o
f le
ngth
n:
H
e c
an
go
thro
ugh t
he s
trin
g b
it b
y b
it:
10
01
00
11
00
00
00
00
4
10
00
00
00
5
11
00
00
00
4
...
10
01
00
11
8
O(n)
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e M
as
ter
Min
d P
rob
lem
: O
pti
mal S
tra
teg
ies (
1/2
)
P
aultr
ies t
o fin
d C
aro
le’s
bin
ary
str
ing o
f le
ngth
n:
H
e c
an
go
thro
ugh t
he s
trin
g b
it b
y b
it:
10
01
00
11
00
00
00
00
4
10
00
00
00
5
11
00
00
00
4
...
10
01
00
11
8
O(n)
Can w
e d
o
even
bett
er?
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e M
as
ter
Min
d P
rob
lem
: O
pti
mal S
tra
teg
ies (
2/2
)
11
01
00
11
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e M
as
ter
Min
d P
rob
lem
: O
pti
mal S
tra
teg
ies (
2/2
)
11
01
00
11
11
10
10
01
01
01
00
00
00
01
11
11
1 2 c n
log
n
4 5 4
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Op
tim
al S
tra
teg
ies (
2/2
)
11
01
00
11
11
10
10
01
01
01
00
00
00
01
11
11
1 2 c n
log
n
4 5 4
34
..
4.
5
65
..
5.
2
34
..
4.
5
00
00
00
00
00
00
00
01 1
10
10
01
1
00
...
1
11
...
1
01
...
0
11
11
11
11
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Op
tim
al S
tra
teg
ies (
2/2
)
11
01
00
11
11
10
10
01
01
01
00
00
00
01
11
11
1 2 c n
log
n
4 5 4
34
..
4.
5
65
..
5.
2
34
..
4.
5
00
00
00
00
00
00
00
01 1
10
10
01
1
00
...
1
11
...
1
01
...
0
11
11
11
11
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Op
tim
al S
tra
teg
ies (
2/2
)
11
01
00
11
11
10
10
01
01
01
00
00
00
01
11
11
1 2 c n
log
n
4 5 4
34
..
4.
5
65
..
5.
2
34
..
4.
5
00
00
00
00
00
00
00
01 1
10
10
01
1
00
...
1
11
...
1
01
...
0
11
11
11
11
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
34
..
4
00
00
00
00
Op
tim
al S
tra
teg
ies (
2/2
)
11
01
00
11
11
10
10
01
01
01
00
00
00
01
11
11
1 2 c n
log
n
4 5 4
.5
65
..
5.
2
34
..
4.
5
00
00
00
01 1
10
10
01
1
00
...
1
11
...
1
01
...
0
11
11
11
11
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
34
..
4
00
00
00
00
Op
tim
al S
tra
teg
ies (
2/2
)
11
01
00
11
11
10
10
01
01
01
00
00
00
01
11
11
1 2 c n
log
n
4 5 4
.5
65
..
5.
2
34
..
4.
5
00
00
00
01 1
10
10
01
1
00
...
1
11
...
1
01
...
0
11
11
11
11
Fitn
ess e
limin
ation
tech
niq
ue
[An
il/W
ieg
and
09
], s
ee
als
o [D
./Jo
ha
nn
se
n/K
ötz
ing
/Le
hre
/Wag
ner/
W. 1
1]
O(
n
logn)
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
34
..
4
00
00
00
00
Op
tim
al S
tra
teg
ies (
2/2
)
11
01
00
11
11
10
10
01
01
01
00
00
00
01
11
11
1 2 c n
log
n
4 5 4
.5
65
..
5.
2
34
..
4.
5
00
00
00
01 1
10
10
01
1
00
...
1
11
...
1
01
...
0
11
11
11
11
Fitn
ess e
limin
ation
tech
niq
ue
[An
il/W
ieg
and
09
], s
ee
als
o [D
./Jo
ha
nn
se
n/K
ötz
ing
/Le
hre
/Wag
ner/
W. 1
1]
Θ(
n
logn)
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Inte
rme
dia
te S
um
ma
ry
W
ant to
un
de
rsta
nd t
rac
tab
ilit
y o
f a
pro
ble
mfo
r ge
nera
l-
pu
rpose (
rand
om
ized
) se
arc
h h
euristics
Q
uery
co
mp
lexit
y a
s s
uch is n
ot a
suff
icie
nt
me
asu
re:
Maste
rmin
d p
roble
m
11
01
00
11
Que
ry C
om
ple
xity
Θ(
n
logn)
Searc
h H
euristics
Θ(nlogn)
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e R
an
kin
g-B
as
ed
Bla
ck
-Bo
x M
od
el
O
bse
rva
tio
n:
ma
ny r
ando
miz
ed s
ea
rch h
eu
ristics u
se f
itness
valu
es
only
to c
om
pa
re
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
...
Th
e R
an
kin
g-B
as
ed
Bla
ck
-Bo
x M
od
el
O
bse
rva
tio
n:
ma
ny r
ando
miz
ed s
ea
rch h
eu
ristics u
se f
itness
valu
es
only
to c
om
pa
re
(1+
1)
EA
RL
S
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
...
Th
e R
an
kin
g-B
as
ed
Bla
ck
-Bo
x M
od
el
O
bse
rva
tio
n:
ma
ny r
ando
miz
ed s
ea
rch h
eu
ristics u
se f
itness
valu
es
only
to c
om
pa
re
RL
S(1
+1
) E
A
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e R
an
kin
g-B
as
ed
Bla
ck
-Bo
x M
od
el
Does n
ot
reveal absolu
te f
itn
ess v
alu
es:
Alg
orith
m
Bla
ck-B
ox
= “
Ora
cle
”
f fff
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e R
an
kin
g-B
as
ed
Bla
ck
-Bo
x M
od
el
Does n
ot
reveal absolu
te f
itn
ess v
alu
es:
Alg
orith
m
Bla
ck-B
ox
= “
Ora
cle
”
x xxx
f fff
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e R
an
kin
g-B
as
ed
Bla
ck
-Bo
x M
od
el
Does n
ot
reveal absolu
te f
itn
ess v
alu
es:
Alg
orith
m
Bla
ck-B
ox
= “
Ora
cle
”
Ran
k 1
Ran
k 2
Ran
k 2
x xxx
x xxx
f fff
x xxx x xxx
Ran
k 4x xxx
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e R
an
kin
g-B
as
ed
Bla
ck
-Bo
x M
od
el
Alg
orith
m
Bla
ck-B
ox
= “
Ora
cle
”
x xxx
g ggg(f fff
(x xxx))
f fff
Equiv
ale
nt
form
ula
tio
n:Le
t be a
str
ictly m
ono
ton
e f
unction
g:R→R
g gggg(f
(x))
=1
27
R
ank 1
g(f
(x))
=1
25
R
ank 2
g(f
(x))
=1
25
R
ank 2
g(f
(x))
=2
7
R
ank 4
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
W
ant to
un
de
rsta
nd t
rac
tab
ilit
y o
f a
pro
ble
m
for
gene
ral-p
urp
ose (
rand
om
ized
) se
arc
h h
euristics
Q
uery
co
mp
lexit
y a
s s
uch is n
ot a
suff
icie
nt
me
asu
re
(M
any)
Ra
ndo
miz
ed s
ea
rch h
eu
ristics d
o s
ele
ctio
n b
ased o
n
rela
tive f
itn
ess
va
lues
only
, n
ot
on a
bsolu
te v
alu
es:
Ran
kin
g-B
ase
d B
lac
k-B
ox
Mo
de
l
Inte
rme
dia
te S
um
ma
ry
Maste
rmin
d p
roble
m
11
01
00
11
Does it
help
?
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e R
an
kin
g-B
as
ed
BB
C o
f M
as
term
ind
is Θ ΘΘΘ
(n nnn/ lo
g n nnn
)
Maste
rmin
d p
roble
m
11
01
00
11
Rankin
g-B
ased
Que
ry C
om
ple
xity
Θ(
n
logn)
Searc
h H
euristics
Θ(nlogn)
Cla
ssic
al
Que
ry C
om
ple
xity
Θ(
n
logn)
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
C
aro
lech
ooses a
bin
ary
str
ing o
f le
ngth
na
nd a
perm
uta
tion σ
P
aulw
ants
to fin
d it.
He m
ay a
sk a
ny s
trin
g o
f le
ngth
n:
C
aro
leco
mp
ute
s th
e w
eig
hte
d f
itn
ess v
alu
e:
“P
aul, y
ou
r str
ing h
as a
sco
re o
f 336
(=
24+
28+
26)”
Ex
am
ple
: B
ina
ryV
alu
e//W
eig
hte
d M
as
term
ind
11
01
00
11
10
10
10
10
24
22
21
23
25
28
26
27
σ=
(4 2
1 3
5 8
6 7
)
10
10
10
10
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Paul can d
o a
bin
ary
sea
rch
(pa
ralle
l fo
r e
ach i≤n
):
Th
e Q
ue
ry C
om
ple
xit
y o
f B
ina
ryV
alu
eis
O(l
ogn
)
11
11
11
11
11
01
00
11
11
11
00
00
11
00
11
00
10
10
10
10
11
01
00
11
24
22
21
23
25
28
26
27
24+
22+
23+
26+
27
24+
22+
23+
25+
28
24+
22+
21
24+
28+
26
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Paul can d
o a
bin
ary
sea
rch
(pa
ralle
l fo
r e
ach i≤n
):
Bin
ary
Se
arc
h n
ot
Po
ss
ible
in
Ra
nk
ing
-Ba
se
d M
od
el
11
11
11
11
11
11
00
00
11
00
11
00
10
10
10
10
11
01
00
11
24
22
21
23
25
28
26
27
28
31
7
-29
29
??
??
??
??
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Paul can d
o a
bin
ary
sea
rch
(pa
ralle
l fo
r e
ach i≤n
):
Bin
ary
Se
arc
h n
ot
Po
ss
ible
in
Ra
nk
ing
-Ba
se
d M
od
el
11
11
11
11
11
11
00
00
11
00
11
00
10
10
10
10
11
01
00
11
24
22
21
23
25
28
26
27
28
31
7
-29
29
In f
act,
RB
BB
C(B
ina
ryV
alu
e)=
Θ(n
)
??
??
??
??
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e R
an
kin
g-B
as
ed
Bla
ck
-Bo
x C
om
ple
xit
y o
f B
inary
Valu
eis
Θ ΘΘΘ(n nnn
)
Lim
ite
d
Le
arn
ing
t=2
If A
lgo
rith
m q
ue
rie
s t
wo
str
ing
s x xxx
an
d y yyy
,
it c
an
le
arn
at
mo
st
1 b
it o
f th
e t
arg
et
str
ing
z zzz.
Exam
ple
:
x =
10000
0
y =
00000
0
g(BVz,σ(x))>g(BVz,σ(y))
⇔
z 1=x1=1
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Th
e R
an
kin
g-B
as
ed
Bla
ck
-Bo
x C
om
ple
xit
y o
f B
inary
Valu
eis
Θ ΘΘΘ(n nnn
)
Lim
ite
d
Le
arn
ing
t ttt=
2
If A
lgo
rith
m q
ue
rie
s t
wo
str
ing
s x xxx
an
d y yyy
,
it c
an
le
arn
at
mo
st
1 b
it o
f th
e t
arg
et
str
ing
z zzz.
Lim
ite
d
Le
arn
ing
g
en
era
l t ttt
If A
lgo
rith
m q
ue
rie
s t ttt
str
ing
s x xxx
1,...
,x xxxt ttt,
it c
an
le
arn
at
mo
st t ttt-
1 b
it o
f th
e t
arg
et
str
ing
z zzz.
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ω ΩΩΩ(n
)
de
term
inis
tic
low
er
bo
un
d
Th
e R
an
kin
g-B
as
ed
Bla
ck
-Bo
x C
om
ple
xit
y o
f B
inary
Valu
eis
Θ ΘΘΘ(n nnn
)
Lim
ite
d
Le
arn
ing
t ttt=
2
If A
lgo
rith
m q
ue
rie
s t
wo
str
ing
s x xxx
an
d y yyy
,
it c
an
le
arn
at
mo
st
1 b
it o
f th
e t
arg
et
str
ing
z zzz.
Lim
ite
d
Le
arn
ing
g
en
era
l t ttt
If A
lgo
rith
m q
ue
rie
s t ttt
str
ing
s x xxx
1,...
,x xxxt ttt,
it c
an
le
arn
at
mo
st t ttt-
1 b
it o
f th
e t
arg
et
str
ing
z zzz.
Th
ere
is
no
de
term
inis
tic
alg
ori
thm
wh
ich
op
tim
ize
s B
ina
ryV
alu
ein
su
bli
ne
ar
tim
e.
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ω ΩΩΩ(n
)
de
term
inis
tic
low
er
bo
un
d
Th
e R
an
kin
g-B
as
ed
Bla
ck
-Bo
x C
om
ple
xit
y o
f B
inary
Valu
eis
Θ ΘΘΘ(n nnn
)
Lim
ite
d
Le
arn
ing
t ttt=
2
If A
lgo
rith
m q
ue
rie
s t
wo
str
ing
s x xxx
an
d y yyy
,
it c
an
le
arn
at
mo
st
1 b
it o
f th
e t
arg
et
str
ing
z zzz.
Lim
ite
d
Le
arn
ing
g
en
era
l t ttt
If A
lgo
rith
m q
ue
rie
s t ttt
str
ing
s x xxx
1,...
,x xxxt ttt,
it c
an
le
arn
at
mo
st t ttt-
1 b
it o
f th
e t
arg
et
str
ing
z zzz.
Th
ere
is
no
de
term
inis
tic
alg
ori
thm
wh
ich
op
tim
ize
s B
ina
ryV
alu
ein
su
bli
ne
ar
tim
e.
Ω ΩΩΩ(n
)
ran
do
miz
ed
lo
wer
bo
un
d
Fo
llo
ws f
rom
de
term
inis
tic
lo
wer
bo
un
d a
nd
Y
ao
’s m
inim
ax
pri
nc
iple
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
W
ant to
un
de
rsta
nd t
racta
bili
ty o
f diffe
rent
pro
ble
ms
for
(rand
om
ized
) sea
rch h
euristics
M
easu
re:
num
be
r o
f fu
nctio
n e
valu
ations
Su
mm
ary
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
W
ant to
un
de
rsta
nd t
racta
bili
ty o
f diffe
rent
pro
ble
ms
for
(rand
om
ized
) sea
rch h
euristics
M
easu
re:
num
be
r o
f fu
nctio
n e
valu
ations
O
ur
main
inte
rest a
re g
oo
d l
ow
er
bo
un
ds
C
lassic
al q
ue
ry c
om
ple
xity: o
fte
n too
weak low
er
boun
ds
Su
mm
ary
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
W
ant to
un
de
rsta
nd t
racta
bili
ty o
f diffe
rent
pro
ble
ms
for
(rand
om
ized
) sea
rch h
euristics
M
easu
re:
num
be
r o
f fu
nctio
n e
valu
ations
O
ur
main
inte
rest a
re g
oo
d l
ow
er
bo
un
ds
C
lassic
al q
ue
ry c
om
ple
xity: o
fte
n too
weak low
er
boun
ds
R
an
kin
g-B
ase
d B
lac
k-B
ox
Mo
de
l:
que
ry c
om
ple
xity m
od
el
only
rela
tive, n
ot
abso
lute
fitness v
alu
es a
re g
iven
Su
mm
ary
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
W
ant to
un
de
rsta
nd t
racta
bili
ty o
f diffe
rent
pro
ble
ms
for
(rand
om
ized
) sea
rch h
euristics
M
easu
re:
num
be
r o
f fu
nctio
n e
valu
ations
O
ur
main
inte
rest a
re g
oo
d l
ow
er
bo
un
ds
C
lassic
al q
ue
ry c
om
ple
xity: o
fte
n too
weak low
er
boun
ds
R
an
kin
g-B
ase
d B
lac
k-B
ox
Mo
de
l:
only
rela
tive, n
ot
abso
lute
fitness v
alu
es a
re g
iven
Su
mm
ary B
inary
Valu
epro
ble
m
11
01
00
11
Rankin
g-B
ased
Query
Com
ple
xity
Searc
h H
euristics
Cla
ssic
al
Query
Com
ple
xity
Θ(nlogn)
Θ(logn)
Θ(n)
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
W
ant to
un
de
rsta
nd t
racta
bili
ty o
f diffe
rent
pro
ble
ms
for
(rand
om
ized
) sea
rch h
euristics
M
easu
re:
num
be
r o
f fu
nctio
n e
valu
ations
O
ur
main
inte
rest a
re g
oo
d l
ow
er
bo
un
ds
C
lassic
al q
ue
ry c
om
ple
xity: o
fte
n too
weak low
er
boun
ds
R
an
kin
g-B
ase
d B
lac
k-B
ox
Mo
de
l:
only
rela
tive, n
ot
abso
lute
fitness v
alu
es a
re g
iven
Su
mm
ary B
inary
Valu
epro
ble
m
11
01
00
11
Rankin
g-B
ased
Query
Com
ple
xity
Searc
h H
euristics
Cla
ssic
al
Query
Com
ple
xity
Θ(nlogn)
Maste
rmin
d p
roble
m
11
01
00
11
Rankin
g-B
ased
Query
Com
ple
xity
Searc
h H
euristics
Cla
ssic
al
Query
Com
ple
xity
Θ(nlogn)
Θ(
n
logn)
Θ(
n
logn)
Θ(logn)
Θ(n)
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ple
nty
!
Fu
ture
Wo
rk
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ple
nty
!
O
the
r bla
ck-b
ox m
od
els
:
re
str
icte
d m
em
ory
models
unbia
se
d s
am
plin
g s
trate
gie
s
com
bin
atio
ns th
ere
of
...
Fu
ture
Wo
rk
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ple
nty
!
O
the
r bla
ck-b
ox m
od
els
:
re
str
icte
d m
em
ory
models
unbia
se
d s
am
plin
g s
trate
gie
s
com
bin
atio
ns th
ere
of
...
C
om
bin
ato
rial p
roble
ms:
M
ST
, S
SS
P p
roble
ms
pa
rtitio
n p
roble
m
...
...
Fu
ture
Wo
rk
B.
Doerr
/C. W
inzen: R
ankin
g-B
ased B
lack-B
ox M
odel
CS
R,
June 1
4,
2011
Ple
nty
!
O
the
r bla
ck-b
ox m
od
els
:
re
str
icte
d m
em
ory
models
unbia
se
d s
am
plin
g s
trate
gie
s
com
bin
atio
ns th
ere
of
...
C
om
bin
ato
rial p
roble
ms:
M
ST
, S
SS
P p
roble
ms
pa
rtitio
n p
roble
m
...
...
Fu
ture
Wo
rk
10
47
36
25
1
98
How
oft
en
do
yo
u n
eed
to
use
th
e
bala
nce
to
fin
d a
pe
rfect
pa
rtitio
n
of
the
balls
?
Alm
ost
equ
ivale
nt
pro
ble
m:
nd
isting
uis
ha
ble
balls
of
unkno
wn
we
igh
t