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