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Lecture 3

Uninformed Search

http://www.darpa.mil/grandchallenge/
http://www.darpa.mil/grandchallenge/http://www.darpa.mil/grandchallenge/

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Complexity Recap app.!"

#We often want to characterize algorithms independent of the implementation.

#This algorithm took 1 hour and 43 seconds on my laptop.Is not very useful, because tomorrow computers are faster.

#Better is: This algorithm takes O(nlog(n)) time to run and O(n) to storbecause this statement is abstracts away from irrelevant deta

Timen! " #fn!! means:Timen! $ constant fn! for n%n& for some n&

'pacen! idem.

n is some variable which characterizes the size of the problem,e.g. number of data(points, number of dimensions, branching(of search tree, etc.

#Worst caseanalysis versus average caseanalyis

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Uninformed $earch$trategie$

Uninformed: %hile $earching you ha&e noclue whether one non'goal $tate i$ (etterthan any other. )our $earch i$ (lind.

*ariou$ (lind $trategie$: +readth',r$t $earch

Uniform'co$t $earch

-epth',r$t $earch terati&e deepening $earch

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+readth',r$t $earch

0xpand $hallowe$t unexpanded node

mplementation: fringei$ a ,r$t'in',r$t'out " ueue4

i.e.4 new $ucce$$or$ go at end of theueue.

$ ! a goal $tate5

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+readth',r$t $earch


mplementation: fringei$ a ueue4 i.e.4 new

$ucce$$or$ go at end

0xpand:fringe 7 8+4C9

$ + a goal $tate5

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+readth',r$t $earch


mplementation: fringei$ a ueue4 i.e.4 new $ucce$$or$

go at end

0xpand:fringe78C4-409

$ C a goal $tate5

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+readth',r$t $earch


mplementation: fringei$ a ueue4 i.e.4 new $ucce$$or$

go at end

0xpand:fringe78-4044

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0xample

+S

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?ropertie$ of (readth',r$t$earch

Complete5)e$ it alway$ reache$ goal if bi$,nite"

@ime51+b+b2+b3AB AbdA bd+1-b)" 7 (dA1"

thi$ i$ the num(er of node$ we generate" Space5O(bd+1)eep$ e&ery node in memory4

either in fringe or on a path to fringe". ptimal5)e$ if we guarantee that deeper

$olution$ are le$$ optimal4 e.g. $tep'co$t71".

Spacei$ the (igger pro(lem more than time"

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1D

Uniform'co$t $earch

+readth',r$t i$ only optimal if $tep co$t$ i$ increa$ingwith depth e.g. con$tant". Can we guaranteeoptimality for any $tep co$t5

Uniform'co$t Search: 0xpand node with

$malle$t path co$tgn".

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Uniform'co$t $earch

mplementation: fringe7 ueue ordered (y path co$t0ui&alent to (readth',r$t if all $tep co$t$ all eual.

Complete5)e$4 if $tep co$t E Fotherwi$e it can get $tuc in in,nite loop$"

@ime5G of node$ withpath cost H co$t of optimal

$olution.

Space5G of node$ on path$ with path co$t H co$t ofoptimal

$olution.

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-epth',r$t $earch

0xpand deepestunexpanded node

mplementation: fringe 7 La$t n ir$t ut L?" ueue4 i.e.4 put

$ucce$$or$ at front

$ ! a goal $tate5

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-epth',r$t $earch

0xpand deepe$t unexpanded node

mplementation: fringe 7 L ueue4 i.e.4 put $ucce$$or$ at front

ueue78+4C9

$ + a goal $tate5

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-epth',r$t $earch



ueue78-404C9

$ - 7 goal $tate5

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-epth',r$t $earch



ueue78I4404C9

$ I 7 goal $tate5

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-epth',r$t $earch



ueue78404C9

$ 7 goal $tate5

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1;

-epth',r$t $earch



ueue7804C9

$ 0 7 goal $tate5

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1=

-epth',r$t $earch



ueue78J4K4C9

$ J 7 goal $tate5

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1>

-epth',r$t $earch



ueue78K4C9

$ K 7 goal $tate5

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2D

-epth',r$t $earch



ueue78C9

$ C 7 goal $tate5

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-epth',r$t $earch



ueue784

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-epth',r$t $earch



ueue78L44

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-epth',r$t $earch



ueue784

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0xample -S

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?ropertie$ of depth',r$t$earch

Complete5Mo: fail$ in in,nite'depth $pace$

Can modify to a&oid repeated $tate$ along path

@ime5O(bm)with m7maximum depth

terri(le if mi$ much larger than d (ut if $olution$ are den$e4 may (e much fa$ter than

(readth',r$t

Space5O(bm), i.e.4 linear $paceN we only need to

remem(er a $ingle path A expanded unexplored node$" ptimal5Mo t may ,nd a non'optimal goal ,r$t"

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terati&e deepening $earch

#@o a&oid the in,nite depth pro(lem of -S4 we candecide to only $earch until depth L4 i.e. we donOt expand (eyond de-epth'Limited Search

#%hat of $olution i$ deeper than L5 ncrea$e L iterati&ely. terati&e -eepening Search

#!$ we $hall $ee: thi$ inherit$ the memory ad&antage of -epth'ir$t$earch.

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2;

terati&e deepening $earchL7D

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2=

terati&e deepening $earchL71

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2>

terati&e deepening $earchlL72

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3D

terati&e deepening $earchlL73

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terati&e deepening $earch

Mum(er of node$ generated in a depth'limited $earch todepth dwith (ranching factor b:

NDLS b!+ b1+ b2+ " + bd-2+ bd-1+ bd

Mum(er of node$ generated in an iterati&e deepening$earch to depth dwith (ranching factor b:M-S7 dA1"(DA d (1A d'1"(2A B A 3(d'2A2(d'1A 1(d

7

or b 1!4 d #4 M-LS 7 1 A 1D A 1DD A 14DDD A 1D4DDD A 1DD4DDD 7 1114111 M-S7 A 6D A DD A 34DDD A 2D4DDD A 1DD4DDD 7 12346D M+S 7 ............................................................................................ 7 1411141DD

1( ) ( )d dO b O b +

+S

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?ropertie$ of iterati&e deepening$earch

Complete5)e$

@ime5 (d+1)b!+ d b1+ (d-1)b2+ " +

bd O(bd) Space5O(bd)

ptimal5)e$4 if $tep co$t 7 1 or

increa$ing function of depth.

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0xample -S

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+idirectional Search

dea $imultaneou$ly $earch forward from S and (acward$

from N 7 324==D $tate$

S

B

C

S

B C

SC B S

State SpaceExample of a Search Tree

$u(optimal (ut practical

optimal (ut memory inecient

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Summary

?ro(lem formulation u$ually reuire$ a($tractingaway real'world detail$ to de,ne a $tate $pace thatcan fea$i(ly (e explored

*ariety of uninformed $earch $trategie$

terati&e deepening $earch u$e$ only linear $paceand not much more time than other uninformedalgorithm$

http://www.c$.rmit.edu.au/!'Search/?roduct/http://aima.c$.(ereley.edu/demo$.html for more demo$"
http://www.cs.rmit.edu.au/AI-Search/Product/http://aima.cs.berkeley.edu/demos.htmlhttp://aima.cs.berkeley.edu/demos.htmlhttp://www.cs.rmit.edu.au/AI-Search/Product/

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