ai05_04_heursearch notes

7/18/2019 AI05_04_HeurSearch notes

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Heuristic search, page 1CSI 4106, Winter 2005

Heuristic searchPointsDefinitionsBest-first searchHill climbingProblems with hill climbing An example: local and globalheuristic functions

Simulated annealing

The A* procedureeans-ends anal!sis


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Definitions

Heuristics "#ree$ heuriskein % find& disco'er(: )the stud!of the methods and rules of disco'er! and in'ention)

+e use our $nowledge of the problem to consider some "not all( successors of the current state "preferabl! ,ustone& as with an oracle( This means pruning the statespace& gaining speed& but perhaps missing the solution.n chess: consider one "apparentl! best( mo'e& ma!be a few -- butnot all possible legal mo'es

.n the tra'elling salesman problem: select one nearest cit!& gi'e upcomplete search "the greed! techni/ue( This gi'es us& in pol!nomialtime& an approximate solution of the inherentl! exponential problem0 itcan be pro'en that the approximation error is bounded


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Definitions "1(

2or heuristic search to wor$& we must be able to ran$the children of a node A heuristic function ta$es a stateand returns a numeric 'alue -- a composite assessmentof this state +e then choose a child with the best score"this could be a maximum or minimum(

A heuristic function can help gain or lose a lot& butfinding the right function is not alwa!s eas!

The 3-pu44le: how man! misplaced tiles5 how man!slots awa! from the correct place5 and so on

+ater ,ugs: 5556hess: no simple counting of pieces is ade/uate


4/28Heuristic search, page 4CSI 4106, Winter 2005

Definitions "7(

The principal gain -- often spectacular -- is thereduction of the state space 2or example& thefull tree for Tic-Tac-Toe has 8 lea'es .f weconsider s!mmetries& the tree becomes sixtimes smaller& but it is still /uite large

+ith a fairl! simple heuristic function we canget the tree down to 9 states " ore on thiswhen we discuss games (

Heuristics can also help speed up exhausti'e &blind search& such as depth-first and breadth-first search



Best-first search

The algorithmselect a heuristic function "e g & distance to the goal(0put the initial node"s( on the open list0repeat

select ;& the best node on the open list0succeed if ; is a goal node0 otherwise put ; on theclosed list and add ;



Hill climbing

This is a greedy algorithm: go as high up as possibleas fast as possible& without loo$ing around too much

The algorithm

select a heuristic function0

set 6& the current node& to the highest-'alued initialnode0

loop

select ;& the highest-'alued child of 60

return 6 if its 'alue is better than the 'alue of ;0otherwise set 6 to ;0


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Problems with hill climbing

= >ocal maximum& or the foothill problem: there is apea$& but it is lower than the highest pea$ in thewhole space

1 The plateau problem: all local mo'es are e/uall!

unpromising& and all pea$s seem far awa!7 The ridge problem: almost e'er! mo'e ta$es us

down

?andom-restart hill climbing is a series of hill-climbing searches with a randoml! selected startnode whene'er the current search gets stuc$

See also simulated annealing -- in a moment


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A hill climbing example

H

G

F

E

D

C

B

A

Goal

state

Initial

state

H

G

F

E

D

C

B

A

H

G

F

E

D

C

B A

Move

1

H

G

F

E

D

C

B A

Move

2a

H

G

F

E

D

C

B A

Move

2b


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A hill climbing example "1(

local heuristic !unction6ount @= for e'er! bloc$ that sits on the correctthing The goal state has the 'alue @3

6ount -= for e'er! bloc$ that sits on an incorrectthing .n the initial state bloc$s 6& D& & 2& #& Hcount @= each Bloc$s A& B count -= each & for thetotal of @9

o'e = gi'es the 'alue @ "A is now on the correctsupport( o'es 1a and 1b both gi'e @9 "B and H

are wrongl! situated( This means we ha'e a localmaximum of @


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glo"al heuristic !unction

6ount @; for e'er! bloc$ that sits on a correctstac$ of ; things The goal state has the 'alue@13

6ount -; for e'er! bloc$ that sits on an incorrectstac$ of ; things That is& there is a largepenalt! for bloc$s tied up in a wrong structure

.n the initial state 6& D& & 2& #& H count -=& -1&-7& -9& -C& - A counts - & for the total of -13

continued


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o'e = gi'es the 'alue -1= "A is now on thecorrect support(

o'e 1a gi'es -= & because 6& D& & 2& #& Hcount -=& -1& -7& -9& -C& -=

o'e 1b gi'es -=C& because 6& D& & 2& #count -=& -1& -7& -9& -C

There is no local maximum

oral: sometimes changing the heuristicfunction is all we need


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Simulated annealing

The intuition for this search method& animpro'ement on hill-climbing& is theprocess of graduall! cooling a li/uidThere is a schedule of )temperatures)&

changing in time ?eaching the )free4ingpoint) "temperature ( stops the process

.nstead of a random restart& we use amore s!stematic method


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Simulated annealing "1(The algorithm& one of the 'ersions

select a heuristic function f0set 6& the current node& to an! initial node0set t& the current time& to 0let schedule"x( be a table of )temperaturesE0

loopt % t @ =0if schedule"t( % & return 60select at random ;& an! child of 60if f";( F f"6(& set 6 to ;& otherwise

set 6 to ; with probabilit! e "f";(-f"6((Gschedule"t( 0


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A few other search ideas

Stochastic beam search:

select w nodes at random0 nodes with higher'alues ha'e a higher probabilit! of selection

#enetic algorithms:generate nodes li$e in stochastic beam search&but from two parents rather than from one

"This topic is worth! of a separate lecture (


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The A* procedureHill-climbing "and its cle'er 'ersions( ma! miss an optimal solutionHere is a search method that ensures optimalit! of the solution

The algorithm$eep a list of partial paths "initiall! root to root& length (0repeat

succeed if the first path P reaches the goal node0

otherwise remo'e path P from the list0extend P in all possible wa!s& add new paths to the list0sort the list b! the sum of two 'alues: the real cost of P till now&and an estimate of the remaining distance0prune the list b! lea'ing onl! the shortest path for each nodereached so far0

untilsuccess or the list of paths becomes empt!0


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The A* procedure "1(

A* re/uires a lower-bound estimate of thedistance of an! node to the goal node Aheuristic that ne'er o'erestimates is alsocalled optimistic or admissible

+e consider three functions with 'alues :

I g"n( is the actual cost of reaching node n&

I h"n( is the actual unknown remaining cost&

I h


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Here is a s$etch! proof of the optimalit! of the goal nodefound b! A* >et m be such a non-optimal goal:

g"m( F g"n $(

and let n & n=& & n$ be a path leading to n $

Since m is a goal& h"m( % and thus h


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The A* procedure "9( A simple search problem: S is the start node& # is the

goal node& the real distances are shown

A lower-bound estimate of the distance to # could beas follows "note that we do not need it for 2(:

A B 6

D 2

#S

1

7

9

9 9

9C C

7 C

A B 6

D 2

#S

C 37 9

8 1== C

= =

=7 C


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The A* procedure "C(Hill-climbinghappens to

succeed here: S

A D

A

B 2

#

S

D

= = 9#2

= = == C 7#1

C 3 8 1 3#5

= 0#0


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The A* procedure " (

A*& step =

S

A D

7@= = 9@8 1


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The A* procedure " (

A*& step 1

S

A D

B D

=7 = =7 1

7@9@C 3 7@C@8 1


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A*& step 7

S

A D

B

6

D

=7 = =7 1

=1 3 = 1

7@9@9@7 9 7@9@C@ =


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A*& step 9

S

A D

B

6

D A

=7 = =7 1

=1 3 = 1 9@C@= =

=9 9 =8 = 9@1@ =


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The A* procedure "= (

A*& step C

S

A D

B

6

2

D A

B

=7 = =7 1

=1 3 = 1 =8 =

=9 9 =8 = =7 =

9@1@C@C 3 9@1@9@7 C


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The A* procedure "==(

A*& step

S

A D

B

6

2

#

D A

B

=7 = =7 1

=1 3 = 1 =8 =

=9 9 =8 = =7 =

= 3 =7 C


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The A* procedure "=1(

The 3-pu44le withoptimistic heuristics

A& & ;: states0m % misplaced0d % depth0f"x( % m"x( @ d"x(

1

2 3

4

5

6

7

8

1

2 3

4

567

8

1

2 3

4

5

6

7

8

1

2 3

4

5

6

7

8

d(x)

=0

d(x)

=1

d(x)

=2

d(x)

=3

m(B)

=5

m(A)

=4

m(C)

=3

m(D)

=5

1

2 3

4

567

8 1

2 3

4

567

8

1

2 3

4

567

8

m(E)

=3

m(F)

=3

m(G)

=4

1

2 3

4

567

8

m(J)

=2

1

2 3

4

567

8

m(K)

=4

m(H)

=3

m(I)

=4

1

2 3

4

56

7

8

12

3

4

567

8

d(x)

=4

1 2 3

4

567

8

m(L)

=1

d(x)

=5

1 2 3

4

567

8

m(M)

=0

1 2 3

4

56

7 8

m(N)

=2


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eans-ends anal!sis

The idea is due to ;ewell and Simon "=8C (: wor$ b!reducing the difference between states& and soapproaching the goal state There are procedures&indexed b! differences& that change states

The #eneral Problem Sol'er algorithm

set 6& the current node& to an! initial node0loop

succeed if the goal state has been reached& otherwisefind the difference between 6 and the goal state0choose a procedure that reduces this difference & appl!it to 6 to produce the new current state0


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eans-ends anal!sis "1( An example: planning a trip

miles Plane Train 6ar Taxi +al$

F = L

= -= L L

= -= L L

K = L

Prere/uisite At plane At train At car At taxi

D

istance

P r o c e d u r e

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