Informed

search

algorit

hms

Chapter

4,Sectio

ns

1–2

Chapte

r4,Sec

tions

1–2

1

Outlin

e

!B

est-

first

sear

ch

!A!

sear

ch

!H

eurist

ics

Chapte

r4,Sec

tions

1–2

2

Rev

iew

:Tre

ese

arc

h

functi

on

Tree-S

earch(p

robl

em,f

ringe

)re

turn

sa

solu

tion

,or

failu

refr

inge

!In

sert(M

ake-N

ode(I

nit

ial-S

tate[p

robl

em])

,fri

nge

)lo

op

do

iffr

inge

isem

ptyth

en

retu

rnfa

ilure

nod

e!

Remove-F

ront(fri

nge

)if

Goal-T

est

[pro

blem

]ap

plie

dto

State(n

ode)

succ

eeds

retu

rnnod

e

frin

ge!

Inse

rtA

ll(E

xpand

(nod

e,pro

blem

),fr

inge

)

Ast

rate

gyis

defin

edby

pick

ing

theord

er

ofnode

expansi

on

Chapte

r4,Sec

tions

1–2

3

Bes

t-firs

tse

arc

h

Idea

:us

ean

eval

uation

func

tion

for

each

node

–es

tim

ate

of“d

esirab

ility

”

"Exp

and

mos

tde

sira

ble

unex

pand

edno

de

Impl

emen

tation

:fr

inge

isa

queu

eso

rted

inde

crea

sing

orde

rof

desira

bilit

y

Spec

ialca

ses:

gree

dyse

arch

A!

sear

ch

Chapte

r4,Sec

tions

1–2

4

Rom

ania

with

step

cost

sin

km

Bucharest

Giurgiu

Urziceni

Hirsova

Eforie

Neamt

Oradea

Zerind

Arad

Timisoara

Lugoj

Mehadia

Dobreta

Craiova

Sibiu

Fagaras

Pitesti

Rim

nicu

Vilc

ea

Vaslui

Iasi

Stra

ight−l

ine

dista

nce

to B

ucha

rest

0160

242

161 77 151

241

366

193

178

253

329 80 199

244

380

226

234

37498

Giurgiu

Urziceni

Hirsova

Eforie

Neamt

Oradea

Zerind

Arad

Timisoara

Lugoj

Mehadia

Dobreta

Craiova

Sibiu

Fagaras

Pitesti

Vaslui

Iasi

Rim

nicu

Vilc

ea

Bucharest

71

75

118

111

70 75120

151

140

99

80

97

10121

1

138

146

85

90

9814292

87

86

Chapte

r4,Sec

tions

1–2

5

Gre

edy

searc

h

Eva

luat

ion

func

tion

h(n

)(h

eurist

ic)

=es

tim

ate

ofco

stfrom

nto

the

clos

est

goal

E.g

.,h

SLD(n

)=

stra

ight

-lin

edi

stan

cefrom

nto

Buc

hare

st

Gre

edy

sear

chex

pand

sth

eno

deth

atappears

tobe

clos

est

togo

al

Chapte

r4,Sec

tions

1–2

6

Gre

edy

searc

hex

am

ple

Arad

366

Chapte

r4,Sec

tions

1–2

7

Gre

edy

searc

hex

am

ple

Zerind

Arad

Sibiu

Timisoara

253

329

374

Chapte

r4,Sec

tions

1–2

8

Gre

edy

searc

hex

am

ple

Rimn

icu V

ilcea

Zerind

Arad

Sibiu

Arad

Fagaras

Oradea

Timisoara

329

374

366

176

380

193

Chapte

r4,Sec

tions

1–2

9

Gre

edy

searc

hex

am

ple

Rimn

icu V

ilcea

Zerind

Arad

Sibiu

Arad

Fagaras

Oradea

Timisoara

Sibiu

Bucharest

329

374

366

380

193

253

0

Chapte

r4,Sec

tions

1–2

10

Pro

per

ties

ofgre

edy

searc

h

Com

plet

e??

Chapte

r4,Sec

tions

1–2

11

Pro

per

ties

ofgre

edy

searc

h

Com

plet

e??

No–

can

get

stuc

kin

loop

s,e.

g.,w

ith

Ora

dea

asgo

al,

Iasi#

Nea

mt#

Iasi#

Nea

mt#

Com

plet

ein

finite

spac

ew

ith

repe

ated

-sta

tech

ecki

ng

Tim

e??

Chapte

r4,Sec

tions

1–2

12

Pro

per

ties

ofgre

edy

searc

h

Com

plet

e??

No–

can

get

stuc

kin

loop

s,e.

g.,

Iasi#

Nea

mt#

Iasi#

Nea

mt#

Com

plet

ein

finite

spac

ew

ith

repe

ated

-sta

tech

ecki

ng

Tim

e??

O(b

m),

but

ago

odhe

uristic

can

give

dram

atic

impr

ovem

ent

Spac

e??

Chapte

r4,Sec

tions

1–2

13

Pro

per

ties

ofgre

edy

searc

h

Com

plet

e??

No–

can

get

stuc

kin

loop

s,e.

g.,

Iasi#

Nea

mt#

Iasi#

Nea

mt#

Com

plet

ein

finite

spac

ew

ith

repe

ated

-sta

tech

ecki

ng

Tim

e??

O(b

m),

but

ago

odhe

uristic

can

give

dram

atic

impr

ovem

ent

Spac

e??

O(b

m)—

keep

sal

lno

des

inm

emor

y

Opt

imal

??

Chapte

r4,Sec

tions

1–2

14

Pro

per

ties

ofgre

edy

searc

h

Com

plet

e??

No–

can

get

stuc

kin

loop

s,e.

g.,

Iasi#

Nea

mt#

Iasi#

Nea

mt#

Com

plet

ein

finite

spac

ew

ith

repe

ated

-sta

tech

ecki

ng

Tim

e??

O(b

m),

but

ago

odhe

uristic

can

give

dram

atic

impr

ovem

ent

Spac

e??

O(b

m)—

keep

sal

lno

des

inm

emor

y

Opt

imal

??N

o

Chapte

r4,Sec

tions

1–2

15

A!se

arc

h

Idea

:av

oid

expa

ndin

gpa

ths

that

are

alre

ady

expe

nsiv

e

Eva

luat

ion

func

tion

f(n

)=

g(n

)+

h(n

)

g(n

)=

cost

sofa

rto

reac

hn

h(n

)=

estim

ated

cost

togo

alfrom

nf(n

)=

estim

ated

tota

lco

stof

path

thro

ugh

nto

goal

A!

sear

chus

esan

adm

issibl

ehe

uristic

i.e.,

h(n

)$

h!(n

)w

here

h!(n

)is

thetr

ue

cost

from

n.

(Also

requ

ire

h(n

)%

0,so

h(G

)=

0fo

ran

ygo

alG

.)

E.g

.,h

SLD(n

)ne

ver

over

estim

ates

the

actu

alro

addi

stan

ce

The

orem

:A!

sear

chis

optim

al

Chapte

r4,Sec

tions

1–2

16

A!se

arc

hex

am

ple

Arad

366=0+366

Chapte

r4,Sec

tions

1–2

17

A!se

arc

hex

am

ple

Zerind

Arad

Sibiu

Timisoara

447=118+329

449=75+374

393=140+253

Chapte

r4,Sec

tions

1–2

18

A!se

arc

hex

am

ple

Zerind

Arad

Sibiu

Arad

Timisoara

Rimn

icu V

ilcea

Fagaras

Oradea

447=118+329

449=75+374

646=280+366

413=220+193

415=239+176671=291+380

Chapte

r4,Sec

tions

1–2

19

A!se

arc

hex

am

ple

Zerind

Arad

Sibiu

Arad

Timisoara

Fagaras

Oradea

447=118+329

449=75+374

646=280+366

415=239+176

Rimn

icu V

ilcea

Craiova

Pitesti

Sibiu

526=366+160

553=300+253

417=317+100

671=291+380

Chapte

r4,Sec

tions

1–2

20

A!se

arc

hex

am

ple

Zerind

Arad

Sibiu

Arad

Timisoara

Sibiu

Bucharest

Rimn

icu V

ilcea

Fagaras

Oradea

Craiova

Pitesti

Sibiu

447=118+329

449=75+374

646=280+366

591=338+253

450=450+0

526=366+160

553=300+253

417=317+100

671=291+380

Chapte

r4,Sec

tions

1–2

21

A!se

arc

hex

am

ple

Zerind

Arad

Sibiu

Arad

Timisoara

Sibiu

Bucharest

Rimn

icu V

ilcea

Fagaras

Oradea

Craiova

Pitesti

Sibiu

Bucharest

Craiova

Rimn

icu V

ilcea

418=418+0

447=118+329

449=75+374

646=280+366

591=338+253

450=450+0

526=366+160

553=300+253

615=455+160

607=414+193

671=291+380

Chapte

r4,Sec

tions

1–2

22

Optim

ality

ofA

!(s

tandard

pro

of)

Supp

ose

som

esu

bopt

imal

goal

G2

has

been

gene

rate

dan

dis

inth

equ

eue.

Letn

bean

unex

pand

edno

deon

ash

orte

stpa

thto

anop

tim

algo

alG

1.

G

n

G2

Start

f(G

2)

=g(G

2)

sinc

eh(G

2)

=0

>g(G

1)

sinc

eG

2is

subop

tim

al

%f(n

)si

nce

his

adm

issi

ble

Sinc

ef(G

2)>

f(n

),A!

will

neve

rse

lect

G2

for

expa

nsio

n

Chapte

r4,Sec

tions

1–2

23

Optim

ality

ofA

!(m

ore

use

ful)

Lem

ma:

A!

expa

nds

node

sin

orde

rof

incr

easing

fva

lue!

Gra

dual

lyad

ds“f

-con

tour

s”of

node

s(c

f.br

eadt

h-fir

stad

dsla

yers

)Con

tour

iha

sal

lno

des

with

f=

fi,

whe

ref i

<f i

+1

O

Z

A T

L M

DC

R

F

P

G

BU

H E

V

I

N

380

400

420

S

Chapte

r4,Sec

tions

1–2

24

Pro

per

ties

ofA

!

Com

plet

e??

Chapte

r4,Sec

tions

1–2

25

Pro

per

ties

ofA

!

Com

plet

e??

Yes

,un

less

ther

ear

ein

finitel

ym

any

node

sw

ith

f$

f(G

)

Tim

e??

Chapte

r4,Sec

tions

1–2

26

Pro

per

ties

ofA

!

Com

plet

e??

Yes

,un

less

ther

ear

ein

finitel

ym

any

node

sw

ith

f$

f(G

)

Tim

e??

Exp

onen

tial

in[rel

ativ

eer

ror

inh&

leng

thof

soln

.]

Spac

e??

Chapte

r4,Sec

tions

1–2

27

Pro

per

ties

ofA

!

Com

plet

e??

Yes

,un

less

ther

ear

ein

finitel

ym

any

node

sw

ith

f$

f(G

)

Tim

e??

Exp

onen

tial

in[rel

ativ

eer

ror

inh&

leng

thof

soln

.]

Spac

e??

Kee

psal

lno

des

inm

emor

y

Opt

imal

??

Chapte

r4,Sec

tions

1–2

28

Pro

per

ties

ofA

!

Com

plet

e??

Yes

,un

less

ther

ear

ein

finitel

ym

any

node

sw

ith

f$

f(G

)

Tim

e??

Exp

onen

tial

in[rel

ativ

eer

ror

inh&

leng

thof

soln

.]

Spac

e??

Kee

psal

lno

des

inm

emor

y

Opt

imal

??Yes

—ca

nnot

expa

ndf

i+1

untilf i

isfin

ishe

d

A!

expa

nds

allno

des

with

f(n

)<

C!

A!

expa

nds

som

eno

des

with

f(n

)=

C!

A!

expa

nds

nono

des

with

f(n

)>

C!

Chapte

r4,Sec

tions

1–2

29

Pro

ofofle

mm

a:

Consi

sten

cy

Ahe

uristic

isco

nsiste

ntif

nc(n,a,n’) h(n’)

h(n) G

n’

h(n

)$

c(n,a

,n" )

+h(n

" )

Ifh

isco

nsiste

nt,we

have

f(n

" )=

g(n

" )+

h(n

" )

=g(n

)+

c(n,a

,n" )

+h(n

" )

%g(n

)+

h(n

)

=f(n

)

I.e.,

f(n

)is

nond

ecre

asin

gal

ong

any

path

.

Chapte

r4,Sec

tions

1–2

30

Adm

issi

ble

heu

rist

ics

E.g

.,fo

rth

e8-

puzz

le:

h1(n

)=

num

ber

ofm

ispl

aced

tile

sh

2(n

)=

tota

lM

anha

ttan

dist

ance

(i.e

.,no

.of

squa

res

from

desire

dlo

cation

ofea

chtile

)

2

Star

t Sta

teG

oal S

tate

5 13

46

785

1

2 3

4 6

7 85

h1(S

)=

??h

2(S

)=

??

Chapte

r4,Sec

tions

1–2

31

Adm

issi

ble

heu

rist

ics

E.g

.,fo

rth

e8-

puzz

le:

h1(n

)=

num

ber

ofm

ispl

aced

tile

sh

2(n

)=

tota

lM

anha

ttan

dist

ance

(i.e

.,no

.of

squa

res

from

desire

dlo

cation

ofea

chtile

)

2

Star

t Sta

teG

oal S

tate

5 13

46

785

1

2 3

4 6

7 85

h1(S

)=

??6

h2(S

)=

??4+

0+3+

3+1+

0+2+

1=

14

Chapte

r4,Sec

tions

1–2

32

Dom

inance

Ifh

2(n

)%

h1(n

)fo

ral

ln

(bot

had

missibl

e)th

enh

2do

min

ates

h1

and

isbe

tter

for

sear

ch

Typ

ical

sear

chco

sts:

d=

14ID

S=

3,47

3,94

1no

des

A!(h

1)

=53

9no

des

A!(h

2)

=11

3no

des

d=

24ID

S'

54,0

00,0

00,0

00no

des

A!(h

1)

=39

,135

node

sA!(h

2)

=1,

641

node

s

Giv

enan

yad

missibl

ehe

uristics

ha,h

b,

h(n

)=

max

(ha(n

),h

b(n))

isal

soad

missibl

ean

ddo

min

ates

ha,h

b

Chapte

r4,Sec

tions

1–2

33

Rel

axed

pro

ble

ms

Adm

issibl

ehe

uristics

can

bede

rive

dfrom

theexact

solu

tion

cost

ofa

rela

xed

vers

ion

ofth

epr

oble

m

Ifth

eru

les

ofth

e8-

puzz

lear

ere

laxe

dso

that

atile

can

mov

eanyw

here

,th

enh

1(n

)gi

ves

the

shor

test

solu

tion

Ifth

eru

les

are

rela

xed

soth

ata

tile

can

mov

eto

any

adja

cent

square

,th

enh

2(n

)gi

ves

the

shor

test

solu

tion

Key

poin

t:th

eop

tim

also

lution

cost

ofa

rela

xed

prob

lem

isno

grea

ter

than

the

optim

also

lution

cost

ofth

ere

alpr

oble

m Chapte

r4,Sec

tions

1–2

34

Rel

axed

pro

ble

ms

contd

.

Wel

l-kn

own

exam

ple:

trav

ellin

gsa

lesp

erso

npr

oble

m(T

SP)

Fin

dth

esh

orte

stto

urvi

siting

allci

ties

exac

tly

once

Min

imum

span

ning

tree

can

beco

mpu

ted

inO

(n2)

and

isa

lower

boun

don

the

shor

test

(ope

n)to

ur

Chapte

r4,Sec

tions

1–2

35

Sum

mary

Heu

rist

icfu

nction

ses

tim

ate

cost

sof

shor

test

path

s

Goo

dhe

uristics

can

dram

atic

ally

redu

cese

arch

cost

Gre

edy

best

-firs

tse

arch

expa

nds

lowes

th

–in

com

plet

ean

dno

tal

way

sop

tim

al

A!

sear

chex

pand

slo

wes

tg

+h

–co

mpl

ete

and

optim

al–

also

optim

ally

e!ci

ent

(up

totie-

brea

ks,fo

rfo

rwar

dse

arch

)

Adm

issibl

ehe

uristics

can

bede

rive

dfrom

exac

tso

lution

ofre

laxe

dpr

oble

ms

Chapte

r4,Sec

tions

1–2

36