Announcements

§ This Friday

§ Project 1 due

§ Talk by Jeniya Tabassum

TweeTIME: A Minimally Supervised Method for Recognizing and Normalizing Time Expressions in Twitter

Recap:Search

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

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

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

UniformCostSearch

§ Strategy:expandlowestpathcost

§ Thegood:UCSiscompleteandoptimal!

§ Thebad:§ Exploresoptionsinevery“direction”§ Noinformationaboutgoallocation

Start Goal

…

c £ 3

c £ 2c £ 1

[Demo:contoursUCSempty(L3D1)][Demo:contoursUCSpacman smallmaze(L3D3)]

VideoofDemoContoursUCSPacman SmallMaze

InformedSearch

SearchHeuristics§ Aheuristicis:

§ Afunctionthatestimates howcloseastateistoagoal§ Designedforaparticularsearchproblem§ Examples:Manhattandistance,Euclideandistancefor

pathing

10

511.2

Example:HeuristicFunction

h(x)

Example:HeuristicFunctionHeuristic:thenumberofthelargestpancakethatisstilloutofplace

43

0

2

3

3

3

4

4

3

4

4

4

h(x)

GreedySearch

§ Strategy:expandanodethatyouthinkisclosesttoagoalstate§ Heuristic:estimateofdistancetonearestgoalforeachstate

§ Acommoncase:§ Best-firsttakesyoustraighttothe(wrong)goal

§ Worst-case:likeabadly-guidedDFS

…b

…b

[Demo:contoursgreedyempty(L3D1)][Demo:contoursgreedypacman smallmaze(L3D4)]

VideoofDemoContoursGreedy(Empty)

VideoofDemoContoursGreedy(Pacman SmallMaze)

A*: CombiningUCSandGreedy

§ Uniform-cost ordersbypathcost,orbackwardcostg(n)§ Greedy ordersbygoalproximity,orforwardcosth(n)

§ A*Search ordersbythesum:f(n)=g(n)+h(n)

S a d

b

Gh=5

h=6

h=2

1

8

11

2

h=6 h=0

c

h=7

3

e h=11

Example:Teg Grenager

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

AdmissibleHeuristics

§ Aheuristich isadmissible (optimistic)if:

whereisthetruecosttoanearestgoal

§ Examples:

§ Comingupwithadmissibleheuristicsismostofwhat’sinvolvedinusingA*inpractice.

415

OptimalityofA*TreeSearch:Blocking

Proof:§ ImagineBisonthefringe§ Someancestorn ofAisonthe

fringe,too(maybeA!)§ Claim:n willbeexpandedbeforeB

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

§ AllancestorsofAexpandbeforeB§ AexpandsbeforeB§ A*searchisoptimal

…

PropertiesofA*

…b

…b

Uniform-Cost A*

UCSvs A*Contours

§ Uniform-costexpandsequallyinall“directions”

§ A*expandsmainlytowardthegoal,butdoeshedgeitsbetstoensureoptimality

Start Goal

Start Goal

[Demo:contoursUCS/greedy/A*empty(L3D1)][Demo:contoursA*pacman smallmaze(L3D5)]

VideoofDemoContours(Empty)-- UCS

VideoofDemoContours(Empty)-- Greedy

VideoofDemoContours(Empty)– A*

VideoofDemoContours(Pacman SmallMaze)– A*

Comparison

Greedy UniformCost A*

A*Applications

§ Videogames§ Pathing /routingproblems§ Resourceplanningproblems§ Robotmotionplanning§ Languageanalysis§ Machinetranslation§ Speechrecognition§ …

[Demo:UCS/A*pacman tinymaze(L3D6,L3D7)][Demo:guessalgorithmEmptyShallow/Deep(L3D8)]

VideoofDemoPacman (TinyMaze)– UCS/A*

CreatingHeuristics

CreatingAdmissibleHeuristics

§ Mostoftheworkinsolvinghardsearchproblemsoptimallyisincomingupwithadmissibleheuristics

§ Often,admissibleheuristicsaresolutionstorelaxedproblems,wherenewactionsareavailable

§ Inadmissibleheuristicsareoftenusefultoo

15366

Example:8Puzzle

§ Whatarethestates?§ Howmanystates?§ Whataretheactions?§ Howmanysuccessorsfromthestartstate?§ Whatshouldthecostsbe?

StartState GoalStateActions

8PuzzleI

§ Heuristic:Numberoftilesmisplaced§ Whyisitadmissible?§ h(start)=§ Thisisarelaxed-problem heuristic

8

Averagenodesexpandedwhentheoptimalpathhas……4steps …8steps …12steps

UCS 112 6,300 3.6x106

TILES 13 39 227

StartState GoalState

StatisticsfromAndrewMoore

8PuzzleII

§ Whatifwehadaneasier8-puzzlewhereanytilecouldslideanydirectionatanytime,ignoringothertiles?

§ TotalManhattandistance

§ Whyisitadmissible?

§ h(start)= 3+1+2+…=18Averagenodesexpandedwhentheoptimalpathhas……4steps …8steps …12steps

TILES 13 39 227MANHATTAN 12 25 73

StartState GoalState

8PuzzleIII

§ Howaboutusingtheactualcost asaheuristic?§ Woulditbeadmissible?§ Wouldwesaveonnodesexpanded?§ What’swrongwithit?

§ WithA*:atrade-offbetweenqualityofestimateandworkpernode§ Asheuristicsgetclosertothetruecost,youwillexpandfewernodesbutusuallydomoreworkpernodetocomputetheheuristicitself

ConsistencyofHeuristics

§ Mainidea:estimatedheuristiccosts≤actualcosts

§ Admissibility:heuristiccost≤actualcosttogoal

h(A)≤ actualcostfromAtoG

§ Consistency:heuristic“arc”cost≤actualcostforeacharc

h(A)– h(C) ≤cost(AtoC)

§ 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