CS343:ArtificialIntelligenceInformedSearch

Prof.ScottNiekum

UniversityofTexasatAustin[TheseslidesbasedononescreatedbyDanKleinandPieterAbbeelforCS188IntrotoAIatUCBerkeley.AllCS188materialsareavailableathttp://ai.berkeley.edu.]

Today

▪ InformedSearch▪Heuristics▪GreedySearch▪ A*Search

▪ GraphSearch

Recap:Search

▪ Searchproblem:▪ States(configurationsoftheworld)▪ Actionsandcosts▪ Successorfunction(worlddynamics)▪ Startstateandgoaltest

▪ Searchtree:▪ Nodes:representplansforreachingstates▪ Planshavecosts(sumofactioncosts)

▪ Searchalgorithm:▪ Systematicallybuildsasearchtree▪ Choosesanorderingofthefringe(unexplorednodes)▪ Optimal:findsleast-costplans

Example:PancakeProblem

Cost:Numberofpancakesflipped

Example:PancakeProblem

3

2

4

3

3

2

2

2

4

Statespacegraphwithcostsasweights

34

3

4

2

GeneralTreeSearch

Action:fliptoptwo Cost:2

Action:flipallfourCost:4Pathtoreachgoal:Flipfour,flipthree

Totalcost:7

TheOneQueue

▪ Allthesesearchalgorithmsarethesameexceptforfringestrategies

▪ Conceptually,allfringesarepriorityqueues(i.e.collectionsofnodeswithattachedpriorities)

▪ Practically,forDFSandBFS,youcanavoidthelog(n)overheadfromanactualpriorityqueue,byusingstacksandqueues

▪ Canevencodeoneimplementationthattakesavariablequeuingobject

UninformedSearch

UniformCostSearch

▪ Strategy:expandlowestpathcost

▪ Thegood:UCSiscompleteandoptimal!

▪ Thebad:▪ Exploresoptionsinevery“direction”▪ Noinformationaboutgoallocation

Start Goal

…

c ≤ 3

c ≤ 2

c ≤ 1

VideoofDemoContoursUCSEmpty

VideoofDemoContoursUCSPacmanSmallMaze

InformedSearch

SearchHeuristics

▪ Aheuristicis:▪ Afunctionthatestimateshowcloseastateistoagoal

▪ Designedforaparticularsearchproblem

▪ Examples:Manhattandistance,Euclideandistanceforpathing

10

5

11.2

Example:HeuristicFunction

h(x)

Example:HeuristicFunction

Heuristic:thenumberofthelargestpancakethatisstilloutofplace

43

0

2

3

3

3

4

4

3

4

4

4

h(x)

GreedySearch

Example:HeuristicFunction

h(x)

GreedySearch

▪ Expandthenodethatseemsclosest…

▪ Whatcangowrong?

GreedySearch

▪ Strategy:expandanodethatyouthinkisclosesttoagoalstate▪ Heuristic:estimateofdistancetonearestgoalforeachstate

▪ Bestcase:▪ Best-firsttakesyoustraighttothenearestgoal

▪ Acommoncase:▪ Suboptimalroutetogoalduetoimperfectheuristic▪ Doesnotleadtonearestgoal

▪ Worst-case:likeabadly-guidedDFS

…b

…b

VideoofDemoContoursGreedy(Empty)

VideoofDemoContoursGreedy(PacmanSmallMaze)

A*Search

A*Search

UCS Greedy

A*

▪ Uniform-costordersbypathcost,orbackwardcostg(n)▪ Greedyordersbygoalproximity,orforwardcosth(n)

▪ A*Searchordersbythesum:f(n)=g(n)+h(n)

CombiningUCSandGreedy

S a d

b

Gh=5

h=6

h=2

1

8

11

2

h=6h=0

c

h=7

3

e h=11

S

a

b

c

ed

dG

G

g=0h=6

g=1h=5

g=2h=6

g=3h=7

g=4h=2

g=6h=0

g=9h=1

g=10h=2

g=12h=0

WhenshouldA*terminate?

▪ Shouldwestopwhenweenqueueagoal?

S

B

A

G

2

3

2

2h=1

h=2

h=0h=3

▪ No:onlystopwhenweexpandagoal

IsA*Optimal?

▪ WhatwillA*dohere?

▪ Whatwentwrong?

▪ Actualbadgoalcost<estimatedgoodgoalcost

▪ Weneedestimatestobelessthanactualcosts!

A

GS

1 3

h=6

h=0

5h=7

Idea:Admissibility

Inadmissible(pessimistic)heuristicsbreakoptimalitybytrappinggoodplansonthefringe

Admissible(optimistic)heuristicscanstillhelptodelaytheevaluationofbadplans,butnever

overestimatethetruecosts

AdmissibleHeuristics

▪ Aheuristichisadmissible(optimistic)if:

whereisthetruecosttoanearestgoal

▪ Examples:

▪ Comingupwithadmissibleheuristicsismostofwhat’sinvolvedinusingA*inpractice.

415

OptimalityofA*TreeSearch

OptimalityofA*TreeSearch

Assume:

▪ Aisanoptimalgoalnode

▪ Bisasuboptimalgoalnode

▪ hisadmissible

Claim:

▪ AwillexitthefringebeforeB

…

OptimalityofA*TreeSearch:Blocking

Proof:

▪ ImagineBisonthefringe

▪ SomeancestornofAisonthefringe,too(maybeA!)

▪ Claim:nwillbeexpandedbeforeB

1. f(n)islessorequaltof(A)

Definitionoff-costAdmissibilityofh

…

h=0atagoal

OptimalityofA*TreeSearch:Blocking

Proof:

▪ ImagineBisonthefringe

▪ SomeancestornofAisonthefringe,too(maybeA!)

▪ Claim:nwillbeexpandedbeforeB

1. f(n)islessorequaltof(A)2. f(A)islessthanf(B)

Bissuboptimalh=0atagoal

…

OptimalityofA*TreeSearch:Blocking

Proof:

▪ ImagineBisonthefringe

▪ SomeancestornofAisonthefringe,too(maybeA!)

▪ Claim:nwillbeexpandedbeforeB

1. f(n)islessorequaltof(A)2. f(A)islessthanf(B)3. nexpandsbeforeB

▪ AllancestorsofAexpandbeforeB

▪ AexpandsbeforeB

▪ A*searchisoptimal

…

PropertiesofA*

PropertiesofA*

…b

…b

Uniform-Cost A*

UCSvsA*Contours

▪ Uniform-costexpandsequallyinall“directions”

▪ A*expandsmainlytowardthegoal,butdoeshedgeitsbetstoensureoptimality

Start Goal

Start Goal

VideoofDemoContours(Empty)--UCS

VideoofDemoContours(Empty)--Greedy

VideoofDemoContours(Empty)–A*

Pacman-A*

Pacman-Greedy

Pacman-UCS

Comparison

Greedy UniformCost A*

Guessalgorithm(DFS/BFS/UCS/Greedy/A*)

Guessalgorithm(DFS/BFS/UCS/Greedy/A*)

Guessalgorithm(DFS/BFS/UCS/Greedy/A*)

Guessalgorithm(DFS/BFS/UCS/Greedy/A*)

Guessalgorithm(DFS/BFS/UCS/Greedy/A*)

A*Applications

▪ Videogames

▪ Pathing/routingproblems

▪ Resourceplanningproblems

▪ Robotmotionplanning

▪ Languageanalysis▪ Machinetranslation

▪ Speechrecognition▪ …

CreatingAdmissibleHeuristics

▪ Mostoftheworkinsolvinghardsearchproblemsoptimallyisincomingupwithadmissibleheuristics

▪ Often,admissibleheuristicsaresolutionstorelaxedproblems,wherenewactionsareavailable

▪ Inadmissibleheuristicsareoftenusefultoo

15366

Example:8Puzzle

▪ Whatarethestates?▪ Howmanystates?▪ Whataretheactions?▪ Howmanysuccessorsfromthestartstate?▪ Whatshouldthecostsbe?

StartState GoalStateActions

8PuzzleI

▪ Heuristic:Numberoftilesmisplaced

▪ Whyisitadmissible?

▪ h(start)=

▪ Thisisarelaxed-problemheuristic

8

Averagenodesexpandedwhentheoptimalpathhas…

…4steps …8steps …12steps

UCS 112 6,300 3.6x106

TILES 13 39 227

StartState GoalState

8PuzzleII

▪ Whatifwehadaneasier8-puzzlewhereanytilecouldslideanydirectionatanytime,ignoringothertiles?

▪ TotalManhattandistance

▪ Whyisitadmissible?

▪ h(start)= 3+1+2+…=18

Averagenodesexpandedwhentheoptimalpathhas…

…4steps …8steps …12steps

TILES 13 39 227

MANHATTAN 12 25 73

StartState GoalState

8PuzzleIII

▪ Howaboutusingtheactualcostasaheuristic?▪ Woulditbeadmissible?▪ Wouldwesaveonnodesexpanded?▪ What’swrongwithit?

▪ WithA*:atrade-offbetweenqualityofestimateandworkpernode▪ Asheuristicsgetclosertothetruecost,youwillexpandfewernodesbutusuallydomoreworkpernodetocomputetheheuristicitself

TrivialHeuristics,Dominance

▪ Dominance:ha≥hcif

▪ Heuristicsformasemi-lattice:▪ Maxofadmissibleheuristicsisadmissible

▪ Trivialheuristics▪ Bottomoflatticeisthezeroheuristic(what

doesthisgiveus?)▪ Topoflatticeistheexactheuristic

GraphSearch

▪ Failuretodetectrepeatedstatescancauseexponentiallymorework.

SearchTreeStateGraph

TreeSearch:ExtraWork!

GraphSearch

▪ InBFS,forexample,weshouldn’tbotherexpandingthecirclednodes(why?)

S

a

b

d p

a

c

e

p

h

f

r

q

q c G

a

qe

p

h

f

r

q

q c G

a

GraphSearch

▪ Idea:neverexpandastatetwice

▪ Howtoimplement:

▪ Treesearch+setofexpandedstates(“closedset”)▪ Expandthesearchtreenode-by-node,but…▪ Beforeexpandinganode,checktomakesureitsstatehasneverbeen

expandedbefore▪ Ifnotnew,skipit,ifnewaddtoclosedset

▪ Important:storetheclosedsetasaset,notalist

▪ Cangraphsearchwreckcompleteness?Why/whynot?

▪ Howaboutoptimality?

A*GraphSearchGoneWrong?

S

A

B

C

G

1

1

1

23

h=2

h=1

h=4

h=1

h=0

S(0+2)

A(1+4) B(1+1)

C(2+1)

G(5+0)

C(3+1)

G(6+0)

Statespacegraph Searchtree

ConsistencyofHeuristics

▪ Mainidea:estimatedheuristiccosts≤actualcosts

▪ Admissibility:heuristiccost≤actualcosttogoal

h(A)≤actualcostfromAtoG

▪ Consistency:heuristic“arc”cost≤actualcostforeacharc

h(A)–h(C)≤cost(AtoC)

i.e.ifthetruecostofanedgefromAtoCisX,thentheh-valueshouldnot

decreasebymorethanXbetweenAandC.

▪ Consequencesofconsistency:

▪ Thefvaluealongapathneverdecreases

h(A)≤cost(AtoC)+h(C)

▪ A*graphsearchisoptimal

3

A

C

G

h=4 h=11

h=2

OptimalityofA*GraphSearch

▪ Sketch:considerwhatA*doeswithaconsistentheuristic:

▪ Fact1:Intreesearch,A*expandsnodesinincreasingtotalfvalue(f-contours)

▪ Fact2:Foreverystates,nodesthatreachsoptimallyareexpandedbeforenodesthatreachssuboptimally

▪ Result:A*graphsearchisoptimal

…

f≤ 3

f≤ 2

f≤ 1

Optimality

▪ Treesearch:▪ A*isoptimalifheuristicisadmissible▪ UCSisaspecialcase(h=0)

▪ Graphsearch:▪ A*optimalifheuristicisconsistent▪ UCSoptimal(h=0isconsistent)

▪ Consistencyimpliesadmissibility

▪ Ingeneral,mostnaturaladmissibleheuristicstendtobeconsistent,especiallyiffromrelaxedproblems

A*:Summary

A*:Summary

▪ A*usesbothbackwardcostsand(estimatesof)forwardcosts

▪ A*isoptimalwithadmissible/consistentheuristics

▪ Heuristicdesigniskey:oftenuserelaxedproblems

TreeSearchPseudo-Code

GraphSearchPseudo-Code