Search' How"do"we"plan"our"holiday?" How"do"we"plan"our"holiday?" Many"problems"can"be"solved"by"search:"

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1 CSC384"Introduc0on"to"Ar0ficial"Intelligence" Search OneofthemostbasictechniquesinAI UnderlyingsubBmoduleinmostAIsystems Cansolvemanyproblemsthathumansarenot goodat(achievingsuperbhumanperformance) Veryusefulasageneralalgorithmictechniquefor solvingmanynonbaiproblems. How"do"we"plan"our"holiday?" Wemusttakeintoaccountvariouspreferencesand constraintstodevelopaschedule. Animportanttechniqueindevelopingsuchascheduleis hypothejcal reasoning. Example:OnholidayinEngland CurrentlyinEdinburgh FlightleavestomorrowfromLondon Needplantogettoyourplane IfItakea6amtrainwherewillIbeat2pm?WillIbesJllable togettotheairportonjme? 1" 2" How"do"we"plan"our"holiday?" Many"problems"can"be"solved"by"search:" ThiskindofhypotheJcalreasoninginvolvesasking whatstatewillibeinauertakingcertainacjons,orauer certainsequencesofevents? FromthiswecanreasonaboutparJcularsequencesof eventsoracjonsoneshouldtrytobringaboutto achieveadesirablestate. SearchisacomputaJonalmethodforcapturinga parjcularversionofthiskindofreasoning. 3" 4"

2 Why"Search?" Successful Successingameplayingprogramsbasedonsearch. ManyotherAIproblemscanbesuccessfullysolvedbysearch. PracJcal Manyproblemsdonthavespecificalgorithmsforsolvingthem. CasJngassearchproblemsisoUentheeasiestwayofsolving them. SearchcanalsobeusefulinapproximaJon(e.g.,localsearchin opjmizajonproblems). ProblemspecificheurisJcsprovidessearchwithawayof exploijngextraknowledge. SomecriJcalaspectsofintelligentbehaviour,e.g., planning,canbenaturallycastassearch. Limita0ons"of"Search" TherearemanydifficultquesJonsthatarenotresolved bysearch.inparjcular,thewholequesjonofhowdoes anintelligentsystemformulatetheproblemitwantsto solveasasearchproblemisnotaddressedbysearch. Searchonlyshowshowtosolvetheproblemoncewe haveitcorrectlyformulated. 5" 6" Search" Represen0ng"a"problem:"The"Formalism" Search"" " FormulaJngaproblemassearchproblem(representaJon) HeurisJcSearch GameBTreeBSearch Toformulateaproblemasasearchproblemweneedthe followingcomponents: 1. STATE"SPACE:"Formulateastatespaceoverwhichwe performsearch.thestatespaceisawayorrepresenjng inacomputerthestatesoftherealproblem. Readings IntroducJon:Chapter UninformedSearch:Chapter3.4 HeurisJcSearch:Chapters3.5, ACTIONS"or"STATE"SPACE"Transi0ons:"FormulateacJons thatallowonetomovebetweendifferentstates.the acjonsreflecttheacjonsonecantakeinthereal problembutoperateonthestatespaceinstead. 7" 8"

3 Represen0ng"a"problem:"The"Formalism" Toformulateaproblemasasearchproblemweneedthe followingcomponents: 3. INITIAL"or"START"STATE"and"GOAL:"IdenJfytheiniJal statethatbestrepresentsthestarjngcondijons,andthe goalorcondijononewantstoachieve. 4. Heuris0cs:"FormulatevariousheurisJcstohelpguidethe searchprocess. The"Formalism" Oncetheproblemhasbeenformulatedasastatespace search,variousalgorithmscanbeujlizedtosolvethe problem. AsoluJontotheproblemwillbeasequenceofacJons/moves thatcantransformyourcurrentstateintoastatewhereyour desiredcondijonholds. 9" 10" Example"1:"Romania"Travel." Example"1." CurrentlyinArad,needtogettoBucharestbytomorrowto catchaflight.whatisthestatespace? Statespace. States:thevariousciJesyoucouldbelocatedin. OurabstracJon:weareignoringthelowleveldetailsof driving,stateswhereyouareontheroadbetweencijes,etc. AcJons:drivebetweenneighboringciJes. IniJalstate:inArad DesiredcondiJon(Goal):beinastatewhereyouarein Bucharest.(HowmanystatessaJsfythiscondiJon?) SoluJonwillbetheroute,thesequenceofciJesto travelthroughtogettobucharest. 11" 12"

4 Example"2." WaterJugs Wehavea3gallon(liter)juganda4gallonjug.Wecanfill eitherjugtothetopfromatap,wecanemptyeitherjug,orwe canpouronejugintotheother(atleastunjltheotherjugis full). States:pairsofnumbers(gal3,gal4)wheregal3isthenumber ofgallonsinthe3gallonjug,andgal4isthenumberofgallons inthe4gallonjug. AcJons:EmptyB3BGallon,EmptyB4BGallon,FillB3BGallon,FillB4B Gallon,PourB3BintoB4,Pour4BintoB3. IniJalstate:Various,e.g.,(0,0) DesiredcondiJon(Goal):Various,e.g.,(0,2)or(*,3)where* meanswedontcare. Example"2." WaterJugs Ifwestartoffwithgal3andgal4asinteger,canonlyreach integervalues. Somevalues,e.g.,(1,2)arenotreachablefromsomeiniJal state,e.g.,(0,0). SomeacJonsarenoBops.Theydonotchangethestate,e.g., (0,0)!EmptyB3BGallon!(0,0) 13" 14" Example"3."The"8WPuzzle" Example"3."The"8WPuzzle" Rule:CanslideaJleintotheblankspot. AlternaJveview:movetheblankspotaround. Statespace. States:ThedifferentconfiguraJonsoftheJles.Howmany differentstates? AcJons:Movingtheblankup,down,leU,right.Canevery acjonbeperformedineverystate? IniJalstate:e.g.,stateshownonpreviousslide. DesiredcondiJon(Goal):beinastatewheretheJlesareall intheposijonsshownonthepreviousslide. SoluJonwillbeasequenceofmovesoftheblank thattransformtheinijalstatetoagoalstate. 15" 16"

5 Example"3."The"8WPuzzle" Althoughthereare9!differentconfiguraJonsofthe Jles(362,880)infactthestatespaceisdividedinto twodisjointparts. Onlywhentheblankisinthemiddleareallfour acjonspossible. OurgoalcondiJonissaJsfiedbyonlyasinglestate. ButonecouldeasilyhaveagoalcondiJonlike The8isintheupperleUhandcorner. HowmanydifferentstatessaJsfythisgoal? More"complex"situa0ons" PerhapsacJonsleadtomulJplestates,e.g.,flipa cointoobtainheadsortails.orwedontknowfor surewhattheinijalstateis(prizeisbehinddoor1, 2,or3).Nowwemightwanttoconsiderhowlikely differentstatesandacjonoutcomesare. ThisleadstoprobabilisJcmodelsofthesearchspace anddifferentalgorithmsforsolvingtheproblem. Laterwewillseesometechniquesforreasoning underuncertainty. 17" 18" Algorithms"for"Search" Algorithms"for"Search" Inputs: aspecifiedinijalstate(aspecificworldstate) asuccessorfuncjons(x)={setofstatesthatcanbereached fromstatexviaasingleacjon}. agoaltestafuncjonthatcanbeappliedtoastateand returnstrueifthestatesajsfiesthegoalcondijon. AnacJoncostfuncJonC(x,a,y)whichdeterminesthecostof movingfromstatextostateyusingacjona.(c(x,a,y)= if adoesnotyieldyfromx).notethatdifferentacjonsmight generatethesamemoveofx!y. Output: asequenceofstatesleadingfromtheinijalstatetoa statesajsfyingthegoaltest. Thesequencemightbe,opJmalincostforsome algorithms,opjmalinlengthforsomealgorithms,come withnoopjmalityguaranteesfromotheralgorithms. 19" 20"

6 Algorithms"for"Search" ObtainingtheacJonsequence. Thesetofsuccessorsofastatexmightarisefromdifferent acjons,e.g., x a y x b z SuccessorfuncJonS(x)yieldsasetofstatesthatcanbereached fromxviaanysingleacjon. Ratherthanjustreturnasetofstates,weannotatethese statesbytheacjonusedtoobtainthem: S(x)={<y,a>,<z,b>} yviaacjona,zviaacjonb. S(x)={<y,a>,<y,b>} yviaacjona,alsoyviaalternajveacjonb. Template"Search"Algorithms" Thesearchspaceconsistsofstates"andacJonsthatmove betweenstates. Apath"inthesearchspaceisasequence"ofstatesconnected byacjons,<s 0,s 1,s 2,,s k >, foreverys i anditssuccessors i+1 theremustexistanacjona i thattransijonss i tos i+1. Alternatelyapathcanbespecifiedby (a)aninijalstates 0,and (b)asequenceofacjonsthatareappliedinturnstarjngfroms 0. Thesearchalgorithmsperformsearchbyexamining alternatepathsofthesearchspace.theobjectsused inthealgorithmarecallednodes eachnode containsapath. 21" 22" Template"Algorithm"for"Search" Template"Algorithm"for"Search" WemaintainasetofFronJernodesalsocalledtheOPENset. ThesenodesarepathsinthesearchspacethatallstartattheiniJal state. IniJallywesetOPEN={<StartState>} Thepath(node)thatstartsandterminatesatthestartstate. AteachstepweselectanodenfromOPEN.Letxbethestate nterminatesat.wecheckifxsajsfiesthegoal,ifnotweadd allextensionsofntoopen(byfindingallstatesins(x)). Search(OPEN, Successors, Goal? ) While(Open not EMPTY) { n = select node from OPEN Curr = terminal state of n If (Goal?(Curr)) return n. OPEN = (OPEN {n}) U s Successors(Curr) <n,s> /* Note OPEN could grow or shrink */ } return FAIL WhendoesOPENgetsmallerinsize? 23" 24"

7 {<Arad>}, {<A,Z>,"<A,T>,"<A,"S>}, {<A,Z>,<A,T>,<A,S,A>,"<A,S,O>,"<A,S,F>,"<A,S,R>} {<A,Z>,<A,T>,<A,S,A>,<A,S,O>,<A,S,R>,<A,S,F,S>,"<A,S,F,B>} {<Arad>}, {<A,Z>,"<A,T>,"<A,"S>}, {<A,Z>,<A,T>,<A,S,A>,"<A,S,O>,"<A,S,F>,"<A,S,R>} {<A,Z>,<A,T>,<A,S,A>,<A,S,O>,<A,S,F>,<A,S,R,S>,"<A,S,R,C>,"<A,S,R,P>} {<A,Z>,<A,T>,<A,S,A>,<A,S,O>,<A,S,F>,"<A,S,R,S>,<A,S,R,C>,<A,S,R,P,R>,<A,S,R,P,C>,<A,S,R,P,B>} Solu0on:"Arad"W>"Sibiu"W>"Fagaras"W>"Bucharest" "Cost:"""""""""140"""+""""99"""+""""211""""="""""450" 25" Solu0on:"Arad"W>"Sibiu"W>"Rimnicu"Vilcea"W>"Pites0"W>"Bucharest" Cost:""""""""""140"""+"""80"""""+""""""""""""""""97"""""""""+""""101"""""=""""418" 26" Selec0on"Rule" TheorderpathsareselectedfromOPENhasacriJcal effectontheoperajonofthesearch: WhetherornotasoluJonisfound ThecostofthesoluJonfound. TheJmeandspacerequiredbythesearch. {<Arad>}, {<A,Z>,<A,T>,<A,S>}, {<A,Z>,<A,T>,<A,S,A>,<A,S,O>,<A,S,F>,<A,S,R>} {<A,Z>,<A,T>,"<A,S,A>,<A,S,O>,<A,S,R>,<A,S,F,S>,<A,S,F,B>}.. cyclescancausenonbterminajon! wedealwiththisissuelater 27" 28"

8 How"to"select"the"next"path"from"OPEN?" AllsearchtechniqueskeepOPENasanorderedset(e.g.,a priorityqueue)andrepeatedlyexecute: IfOPENisempty,terminatewithfailure. GetthenextpathfromOPEN. Ifthepathleadstoagoalstate,terminatewithsuccess. Extendthepath(i.e.generatethesuccessorstatesofthe terminalstateofthepath)andputthenewpathsinopen. Cri0cal"Proper0es"of"Search" Completeness:willthesearchalwaysfindasoluJonifa solujonexists? OpJmality:willthesearchalwaysfindtheleastcost solujon?(whenacjonshavecosts) Timecomplexity:whatisthemaximumnumberofnodes (paths)thancanbeexpandedorgenerated? Spacecomplexity:whatisthemaximumnumberof nodes(paths)thathavetobestoredinmemory? HowdoweorderthepathsonOPEN? 29" 30" Uninformed"Search"Strategies" ThesearestrategiesthatadoptafixedruleforselecJng thenextstatetobeexpanded. TheruledoesnotchangeirrespecJveofthesearch problembeingsolved. Thesestrategiesdonottakeintoaccountanydomain specificinformajonabouttheparjcularsearch problem. Uninformedsearchtechniques: BreadthBFirst,UniformBCost,DepthBFirst,DepthBLimited,andIteraJveB Deepeningsearch BreadthWFirst"Search" 31" 32"

9 BreadthWFirst"Search" Placethenewpathsthatextendthecurrentpathatthe endofopen. WaterJugs.Start=(0,0),Goal=(*,2) Green=NewlyAdded. 1. OPEN={<(0,0)>} 2. OPEN={<(0,0),(3,0)>,"<(0,0),(0,4)>} 3. OPEN={<(0,0),(0,4)>,<(0,0),(3,0),(0,0)>,"" """"""""""""""""<(0,0)",(3,0),(3,4)>,"<(0,0),(3,0),(0,3)">} 4. OPEN={<(0,0),(3,0),(0,0)>,<(0,0),(3,0),(3,4)>, <(0,0),(3,0),(0,3)>,<(0,0),(0,4),(0,0)>,"" """""""""""""""<(0,0),(0,4),(3,4)>,"<(0,0),(0,4),(3,1)>} 33" BreadthWFirst"Search" Level0 Level1 Level2 #2:(3,0) #4:(0,0) #5:(3,4) #6:(0,3) #1:(0,0) #3:(0,4) #7:(0,0) #8:(3,4) #9:(3,1) " Aboveweindicateonlythestatethateachnodesterminatesat.The pathrepresentedbyeachnodeisthepathfromtheroottothatnode. " BreadthBFirstexploresthesearchspacelevelbylevel. 34" BreadthWFirst"Proper0es" Completeness? ThelengthofthepathremovedfromOPENisnonBdecreasing. wereplaceeachexpandednodenwithanextensionofn. Allshorterpathsareexpandedpriorbeforeanylongerpath. Hence,eventuallywemustexamineallpathsoflengthd,and thusfindasolujonifoneexists. OpJmality? BytheabovewillfindshortestlengthsoluJon leastcostsolujon? Notnecessarily:shortestsoluJonnotalwayscheapest solujonifacjonshavevaryingcosts Beadth"first"Solu0on:"Arad"W>"Sibiu"W>"Fagaras"W>"Bucharest" "Cost:"""""""""140"""+""""99"""+""""211""""="""""450" Lowest"cost"Solu0on:"Arad"W>"Sibiu"W>"Rimnicu"Vilcea"W>"Pites0"W>"Bucharest" Cost:""""""""""140"""+"""80"""""+""""""""""""""""97"""""""""+""""101"""""=""""418" 35" 36"

10 BreadthWFirst"Proper0es" MeasuringJmeandspacecomplexity. letbbethemaximumnumberofsuccessorsofany node(maximalbranchingfactor). letdbethedepthoftheshortestsolujon. Rootatdepth0isapathoflength1 Sod=lengthofpathB1 TimeComplexity? BreadthWFirst"Proper0es" SpaceComplexity? O(b d+1 ):Ifgoalnodeislastnodeatleveld,allofthe successorsoftheothernodeswillbeonopenwhenthe goalnodeisexpandedb(b d 1) 1+b+b 2 +b 3 + +b db1 +b d +b(b d 1)=O(b d+1 ) 37" 38" BreadthWFirst"Proper0es" Spacecomplexityisarealproblem. E.g.,letb=10,andsay100,000nodescanbeexpandedper secondandeachnoderequires100bytesofstorage: Depth Nodes Time Memory UniformWCost"Search" millisec. 100bytes sec. 100MB min. 10GB hrs. 100GB TypicallyrunoutofspacebeforewerunoutofJme inmostapplicajons. 39" 40"

11 UniformWCost"Search" KeepOPENorderedbyincreasingcostofthepath. Alwaysexpandtheleastcostpath. IdenJcaltoBreadthfirstifeachacJonhasthesamecost. UniformWCost"Proper0es" Completeness? IfeachtransiJonhascosts ε>0. Thepreviousargumentusedforbreadthfirstsearchholds:the costofthepathrepresentedbyeachnodenchosentobe expandedmustbenonbdecreasing. OpJmality? FindsopJmalsoluJonifeachtransiJonhascost ε>0. Explorespathsinthesearchspaceinincreasingorderofcost. Somustfindminimumcostpathtoagoalbeforefindingany highercostspaths. 41" 42" UniformWCost"Search."Proof"of"Op0mality" UniformWCost"Search."Proof"of"Op0mality" LetusproveOpJmalitymoreformally.Wewillreusethis argumentlateronwhenweexamineheurisjcsearch Lemma"1." Letc(n)bethecostofnodenonOPEN(costofthepath representedbyn).ifn2isexpandedimmediately auern1then c(n1) c(n2). Proof:thereare2cases: a. n2wasonopenwhenn1wasexpanded: We#must#have#c(n1)# #c(n2)#otherwise#n2#would#have#been#selected#for# expansion#rather#than#n1# b. n2wasaddedtoopenwhenn1wasexpanded ##########Now#c(n1)#<#c(n2)#since#the#path#represented#by#n2#extends##the#path# represented#by#n1#and#thus#cost#at#least#ε#more.# 43" 44"

12 UniformWCost"Search."Proof"of"Op0mality" Lemma"2." Whennodenisexpandedeverypathinthesearchspacewith coststrictlylessthanc(n)hasalreadybeenexpanded. Proof: Letnk=<Start,s1,,sk>beapathwithcostlessthanc(n).Letn0= <Start>,n1=<Start,s1>,n2=<Start,s1,s2>,,ni=<Start,s1,,si>,, nk=<start,s1,,sk>.letnibethe#last#node#in#this#sequence#that#has# already#been#expanded#by#search.# So,ni+1mustsJllbeonOPEN:itwasaddedtoopenwhenniwas expanded.alsoc(ni+1) c(nk)<c(n):c(ni+1)isasubpathofnkwehave assumedthatc(nk)is<c(n). ButthenuniformBcostwouldhaveexpandedni+1notn. Soeverynodeniincludingnkmustalreadybeexpanded,i.e.,thislower costpathhasalreadybeenexpanded. 45" UniformWCost"Search."Proof"of"Op0mality" Lemma"3." ThefirstJmeuniformBcostexpandsanoden terminajngatstates,ithasfoundtheminimalcost pathtos(itmightlaterfindotherpathstosbutnone ofthemcanbecheaper). Proof: Allcheaperpathshavealreadybeenexpanded,noneofthemterminatedat S. AllpathsexpandedaUernwillbeatleastasexpensive,sonocheaperpath toscanbefoundlater. So,whenapathtoagoalstateisexpandedthepathmust beopjmal(lowestcost). 46" UniformWCost"Proper0es" TimeandSpaceComplexity? O(b C*/ε )wherec*isthecostoftheopjmalsolujon. Theremaybemanypathswithcost C*:therecanbeasmany asb d pathsoflengthdintheworstcase. PathswithcostlowerthanC*canbeaslongasC*/ε(whyno longer?),somighthaveb C*/ε pathstoexplorebeforefindingan opjmalcostpath. DepthWFirst"Search" 47" 48"

13 DepthWFirst"Search" Placethenewpathsthatextendthecurrentpathatthe frontofopen. WaterJugs.Start=(0,0),Goal=(*,2) Green=NewlyAdded. 1. OPEN={<(0,0)>} 2. OPEN={<(0,0),"(3,0)>,"<(0,0),"(0,4)>} 3. OPEN={<(0,0),(3,0),(0,0)>,"<(0,0),(3,0),(3,4)>,"" """"""""""""""""<(0,0),(3,0),(0,3)>,<(0,4),(0,0)>} 4. OPEN={<(0,0),(3,0),(0,0),(3,0)>,"<(0,0),(3,0),(0,0),(0,4)>"" """""""""""""""<(0,0),(3,0),(3,4)>,<(0,0),(3,0),(0,3)>, <(0,0),(0,4)>} DepthWFirst"Search" Level0 Level1 Level2 Level3 #1:(0,0) #2:(3,0) (0,4) #3:(0,0) (3,4) (0,3) #4:(3,0) (0,4) " Rednodesarebacktrackpoints(thesenodesremainonopen). 49" 50" DepthWFirst"Proper0es" DepthWFirst"Proper0es" Completeness? Infinitepaths?Causeincompleteness! Prunepathswithcycles(duplicatestates) Wegetcompletenessifstatespaceisfinite TimeComplexity? O(b m )wheremisthelengthofthelongestpathinthestate space. Verybadifmismuchlargerthand(shortestpathtoagoal state),butiftherearemanysolujonpathsitcanbemuch fasterthanbreadthfirst.(canbygoodluckbumpintoasolujon quickly). OpJmality? No! 51" 52"

14 DepthWFirst"Proper0es" DepthBFirstBacktrackPoints=unexploredsiblingsof nodesalongcurrentpath. ThesearethenodesthatremainonopenaUerweextracta nodetoexpand. SpaceComplexity? O(bm),linearspace! OnlyexploreasinglepathataJme. OPENonlycontainsthedeepestnodeonthecurrentpath alongwiththebacktrackpoints. AsignificantadvantageofDFS DepthWLimited"Search" 53" 54" Depth"Limited"Search" Depth"Limited"Search" Breadthfirsthasspaceproblems.Depthfirstcanrunoffdowna verylong(orinfinite)path. Depthlimitedsearch PerformdepthfirstsearchbutonlytoapreBspecifieddepthlimitD. THEROOTisatDEPTH0.ROOTisapathoflength1. NonoderepresenJngapathoflengthmorethanD+1isplacedonOPEN. We truncate thesearchbylookingonlyatpathsoflengthd+1orless. Nowinfinitelengthpathsarenotaproblem. ButwillonlyfindasoluJonifasoluJonofDEPTH Dexists. DLS(OPEN, Successors, Goal?) /* Call with OPEN = {<START>} */ WHILE(OPEN not EMPTY) { n= select first node from OPEN Curr = terminal state of n If(Goal?(Curr)) return n If Depth(n) < D //Dont add successors if Depth(n) = D OPEN = (OPEN {n}) U s Successors(Curr) <n,s> Else OPEN = OPEN {n} CutOffOccured = TRUE. } return FAIL WewilluseCutOffOccuredlater. 55" 56"

15 Depth"Limited"Search"Example" Itera0ve"Deepening"Search" 57" 58" Itera0ve"Deepening"Search" Itera0ve"Deepening"Search"Example" SolvetheproblemsofdepthBfirstandbreadthBfirstby extendingdepthlimitedsearch StarJngatdepthlimitL=0,weiteraJvelyincreasethe depthlimit,performingadepthlimitedsearchforeach depthlimit. StopifasoluJonisfound,orifthedepthlimitedsearch failedwithoutcu~ngoffanynodesbecauseofthe depthlimit. Ifnonodeswerecutoff,thesearchexaminedallpathsinthe statespaceandfoundnosolujon!nosolujonexists. 59" 60"

16 Itera0ve"Deepening"Search"Example" Itera0ve"Deepening"Search"Example" 61" 62" Itera0ve"Deepening"Search"Example" Itera0ve"Deepening"Search"Proper0es" Completeness? YesifaminimaldepthsoluJonofdepthdexists. WhathappenswhenthedepthlimitL=d? WhathappenswhenthedepthlimitL<d? TimeComplexity? 63" 64"

17 Itera0ve"Deepening"Search"Proper0es" TimeComplexity (d+1)b 0 +db 1 +(db1)b 2 + +b d =O(b d ) E.g.b=4,d=10 (11)* *4 1 +9* = 1,864, =1,048,576 Mostnodeslieonboomlayer. BFS"can"explore"more"states"than"IDS!" " ForIDS,theJmecomplexityis (d+1)b 0 +db 1 +(db1)b 2 + +b d =O(b d ) " ForBFS,theJmecomplexityis 1+b+b 2 +b 3 + +b d +b(b d 1)=O(b d+1 ) E.g.b=4,d=10 " ForIDS (11)* *4 1 +9* = 1,864,131(statesgenerated) " ForBFS (4 10 1)=5,592,401(statesgenerated) InfactIDScanbemoreefficientthanbreadthfirstsearch:nodesatlimitare notexpanded.bfsmustexpandallnodesunjlitexpandsagoalnode.soa theboomlayeritwilladdmanynodestoopenbeforefindingthegoal node. 65" 66" Itera0ve"Deepening"Search"Proper0es" SpaceComplexity O(bd)SJlllinear! OpJmal? WillfindshortestlengthsoluJonwhichisopJmalifcostsare uniform. Ifcostsarenotuniform,wecanusea cost boundinstead. Path/Cycle"Checking" Onlyexpandpathsofcostlessthanthecostbound. Keeptrackoftheminimumcostunexpandedpathineach depthfirstiterajon,increasethecostboundtothisonthe nextiterajon. Thiscanbemoreexpensive.NeedasmanyiteraJonsofthe searchastherearedisjnctpathcosts. 67" 68"

18 Path"Checking" Example:"Arad"to"Neamt" Ifnkrepresentsthepath<s 0,s 1,,s k >andweexpands k toobtain childc,wehave <S,s 1,,s k,c> Asthepathto c. Pathchecking: Ensurethatthestatecisnotequaltothestatereachedbyany ancestorofcalongthispath. PathsarecheckedinisolaJon! 69" 70" Path"Checking"Example" Cycle"Checking" Arad CycleChecking Keeptrackofallstatespreviouslyexpandedduringthesearch. Timisoara Sibiu Whenweexpandn k toobtainchildc Ensurethatcisnotequaltoanypreviouslyexpandedstate. Arad Lugoj Arad R. Vilcea Fagaras Thisiscalledcyclechecking,ormulJplepathchecking. WhathappenswhenweuJlizethistechniquewithdepthBfirst search? Timisoara Mehadia Sibiu Craiova Pitesti What"happens"to"space"complexity?" Lugoj Drobeta R. Vilcea Drobeta Pitesti 71" 72"

19 Cycle"Checking"Example"(BFS)" Cycle"Checking" Arad Higherspacecomplexity(equaltothespacecomplexityof breadthbfirstsearch. Timisoara Sibiu ThereisanaddiJonalissuewhenwearelookingforan opjmalsolujon" Arad Timisoara Lugoj Lugoj Mehadia Drobeta Arad R. Vilcea Sibiu Craiova R. Vilcea Drobeta Pitesti Fagaras Pitesti WithuniformBcostsearch,wesJllfindanopJmalsoluJon ThefirstJmeuniformBcostexpandsastateithasfoundthe minimalcostpathtoit. Thismeansthatthenodesrejectedsubsequentlybycycle checkingcanthavebeerpaths. Wewillseelaterthatwedontalwayshavethispropertywhen wedoheurisjcsearch. 73" 74" Heuris0c"Search" Heuris0c"Search" (Informed"Search)" Inuninformed"search,wedonttrytoevaluatewhich ofthenodesonopenaremostpromising.wenever lookbahead tothegoal. E.g.,inuniformcostsearchwealwaysexpandthecheapest path.wedontconsiderthecostofge~ngtothegoalfrom theendofthecurrentpath. OUenwehavesomeotherknowledgeaboutthemerit ofnodes,e.g.,goingthewrongdirecjoninromania. 75" 76"

20 Heuris0c"Search" MeritofanOPENnode:differentnoJonsof merit. IfweareconcernedaboutthecostofthesoluJon, wemightwanttoconsiderhowcostlyitistoget tothegoalfromtheterminalstateofthatnode. Ifweareconcernedaboutminimizing computajoninsearchwemightwanttoconsider howeasyitistofindthegoalfromtheterminal stateofthatnode. Wewillfocusonthe costofsolujon nojonof merit. Heuris0c"Search" TheideaistodevelopadomainspecificheurisJc funcjonh(n). h(n)guessesthecostofge~ngtothegoalfromnoden (i.e.,fromtheterminalstateofthepathrepresentedby n). Therearedifferentwaysofguessingthiscostindifferent domains. heurisjcsaredomainspecific. 77" 78" Example:"Euclidean"distance" Heuris0c"Search" Ifh(n 1 )<h(n 2 )thismeansthatweguessthatitis cheapertogettothegoalfromn 1 thanfromn 2. Werequirethat h(n)=0foreverynodenwhoseterminalstatesajsfiesthegoal. ZerocostofachievingthegoalfromnodethatalreadysaJsfies thegoal. Planning a path from Arad to Bucharest, we can utilize the straight line distance from each city to our goal. This lets us plan our trip by picking cities at each time point that minimize the distance to our goal. 79" 80"

21 Using"only"h(n):"Greedy"bestWfirst"search" Greedy"bestWfirst"search"example" Weuseh(n)torankthenodesonOPEN AlwaysexpandnodewithlowesthBvalue. WearegreedilytryingtoachievealowcostsoluJon. However,thismethodignores"the"cost"of"geing"to"n,soitcanbeleadastray exploringnodesthatcostalotbutseemtobeclosetothegoal: step cost = 10 step cost = 100 h(n1) = 20 10" n1 S 100" n3 h(n3) = 10 10" n2 10" 10" Goal 81" 82" A*"search" A * "example" Takeintoaccountthecostofge~ngtothenodeaswellasour esjmateofthecostofge~ngtothegoalfromthenode. DefineanevaluaJonfuncJonf(n) f(n)=g(n)+h(n) g(n)isthecostofthepathrepresentedbynoden h(n)istheheurisjcesjmateofthecostofachievingthegoalfromn. AlwaysexpandthenodewithlowestfBvalueonOPEN. ThefBvalue,f(n)isanesJmateofthecostofge~ngtothegoal viathenode(path)n. I.e.,wefirstfollowthepathnthenwetrytogettothegoal.f(n)esJmates thetotalcostofsuchasolujon. 83" 84"

22 A * "example" A * "example" 85" 86" A * "example" A * "example" 87" 88"

23 A * "example" Proper0es"of"A*"depend"on"condi0ons"on"h(n)" Wewanttoanalyzethebehavioroftheresultantsearch. Completeness,Jmeandspace,opJmality? Toobtainsuchresultswemustputsomefurther condijonsontheheurisjcfuncjonh(n)andthesearch space. 89" 90" Condi0ons"on"h(n):"Admissible" Consistency"(aka"monotonicity)" Wealwaysassumethatc(s1,a,s2) ε>0foranytwo statess1ands2andanyacjona:thecostofany transijonisgreaterthanzeroandcantbearbitrarily small. Leth*(n)bethecost"of"anop0mal"pathfromntoagoal node( ifthereisnopath).thenanadmissible heurisjcsajsfiesthecondijon h(n) h*(n) AstrongercondiJonthanh(n) h*(n). Amonotone/consistentheurisJcsaJsfiesthetriangle inequality:forallnodesn1,n2andforallacjonsa h(n1) C(n1,a,n2)+h(n2) WhereC(n1,a,n2)meansthecostofge~ngfromtheterminal stateofn1totheterminalstateofn2viaacjona. anadmissibleheurisjcneveroverbesjmatesthecosttoreach thegoal,i.e.,itisopjmisjc Henceh(g)=0,foranygoalnodeg Alsoh*(n)= ifthereisnopathfromntoagoalnode NotethattheremightbemorethanonetransiJon(acJon) betweenn1andn2,theinequalitymustholdforallofthem. Monotonicityimpliesadmissibility. (foralln1,n2,a)h(n1) C(n1,a,n2)+h(n2)#(foralln)h(n) h*(n) 91" 92"

24 Consistency"#"Admissible" Intui0on"behind"admissibility" Assume"consistency:"h(n) c(n,a,n2)+h(n2)" "Prove"admissible:"h(n) h*(n) Proof:" Letn!n1!!n*beanOPTIMALpathfromntoagoal(with acjonsa1,a2).notethecostofthispathish*(n),andeachsubpath (ni!!n*)hascostequaltoh*(ni). Ifnopathexistsfromntoagoalthenh*(n)= andh(n) h*(n) Otherwiseproveh(n) h*(n)byinducjononthelengthofthis opjmalpath. Base"Case:"n"="n*" ByourcondiJonsonh,h(n)=0 h(n)*=0 Induc0on"Hypothesis:"h(n1)" h*(n1)"" h(n) c(n,a1,n1)+h(n1) c(n,a1,n1)+h*(n1)=h*(n) 93" h(n) h*(n)meansthatthesearchwon tmissanypromisingpaths. Ifitreallyischeaptogettoagoalvian(i.e.,bothg(n)andh*(n) arelow),thenf(n)=g(n)+h(n)willalsobelow,andthesearch won tignoreninfavorofmoreexpensiveopjons. Thiscanbeformalizedtoshowthatadmissibilityimplies opjmality. MonotonicitygivessomeaddiJonalproperJeswhenitcomesto cyclechecking. 94" Consequences"of"monotonicity" 1. ThefBvaluesofnodesalongapathmustbenonBdecreasing. Let<Start s1 s2 sk>beapath.letnibethesubpath <Start s1 si>: Weclaimthat:f(ni) f(ni+1) Proof f(ni)=c(start ni)+h(ni) c(start ni)+c(ni ni+1)+h(ni+1) c(start ni ni+1)+h(ni+1) g(ni+1)+h(ni+1)=f(ni+1) Consequences"of"monotonicity" 2. Ifn2isexpandedimmediatelyaUern1,then f(n1) f(n2) (thefbvalueofexpandednodesismonotonicnonbdecreasing) Proof:" Ifn2wasonOPENwhenn1wasexpanded, thenf(n1) f(n2)otherwisewewouldhaveexpandedn2. Ifn2wasaddedtoOPENaUern1sexpansion,thenn2extends n1spath.thatis,thepathrepresentedbyn1isaprefixofthe pathrepresentedbyn2.byproperty(1)wehavef(n1) f(n2)as thefbvaluesalongapatharenonbdecreasing. 95" 96"

25 Consequences"of"monotonicity" 3. Corollary:thesequenceoffBvaluesofthenodes expandedbya*isnonbdecreasing.i.e,ifn2is expandedaker(notnecessarilyimmediatelyauer)n1, thenf(n1) f(n2) (thefbvalueofexpandednodesismonotonicnonbdecreasing) Proof: Ifn2wasonOPENwhenn1wasexpanded, thenf(n1) f(n2)otherwisewewouldhaveexpandedn2. Ifn2wasaddedtoOPENaUern1sexpansion,thenletnbe anancestorofn2thatwaspresentwhenn1wasbeing expanded(thiscouldben1itself).wehavef(n1) f(n)since A*chosen1whilenwaspresentonOPEN.Also,sincenis alongthepathton2,byproperty(1)wehavef(n) f(n2).so, wehavef(n1) f(n2). 97" Consequences"of"monotonicity" 4. WhennisexpandedeverypathwithlowerfBvaluehasalready beenexpanded. Proof:AssumebycontradicJonthatthereexistsapath<Start, n0,n1,nib1,ni,ni+1,,nk>withf(nk)<f(n)andniisitslast expandednode. ni+1mustbeonopenwhilenisexpanded,so a)by(1)f(ni+1) f(nk)sincetheyliealongthesamepath. b)sincef(nk)<f(n)sowehavef(ni+1)<f(n) c)by(2)f(n) f(ni+1)becausenisexpandedbeforeni+1. ContradicJonfromb&c! 98" Consequences"of"monotonicity" Consequences"of"monotonicity" 5. WithamonotoneheurisJc,thefirstJmeA*expandsa state,ithasfoundtheminimumcostpathtothatstate. Proof: LetPATH1="<Start,"s0,"s1,","sk,"s>bethe"firstpathtoastates found.wehavef(path1)=c(path1)+h(s). LetPATH2"="<Start,"t0,"t1,","tj,"s>beanotherpathtosfoundlater. wehavef(path2)=c(path2)+h(s). Noteh(s)isdependentonlyonthestates(terminalstateofthe path)itdoesnotdependonhowwegottos. Byproperty(3),f(path1) f(path2) hence:c(path1) c(path2) Complete." Yes,consideraleastcostpathtoagoalnode SoluJonPath=<Start n1 G>withcostc(SoluJonPath).Sinceh(G)= 0,thismeansthatf(SoluJonPath)=cost(SoluJonPath) SinceeachacJonhasacost ε>0,thereareonlyafinitenumberofpaths thathavefbvalue<c(solujonpath).noneofthesepathsleadtoagoalnode sincesolujonpathisaleastcostpathtothegoal. SoeventuallySoluJonPath,orsomeequalcostpathtoagoalmustbe expanded. Time"and"Space"complexity." Whenh(n)=0,forallnhismonotone. A*becomesuniformBcostsearch! Itcanbeshownthatwhenh(n)>0forsomenandsJlladmissible,thenumber ofnodesexpandedcanbenolargerthanuniformbcost. HencethesameboundsasuniformBcostapply.(Theseareworstcasebounds). SJllexponenJalunlesswehaveaverygoodh! Inrealworldproblems,wesomeJmesrunoutofJmeandmemory.IDA*can somejmesbeusedtoaddressmemoryissues,butida*isn tverygoodwhen manycyclesarepresent. 99" 100"

26 Consequences"of"monotonicity" OpJmality " Yes,by(5)thefirstpathtoagoalnodemustbeopJmal. CycleChecking 5. WithamonotoneheurisJc,thefirstJmeA*expandsastate, ithasfoundtheminimumcostpathtothatstate. " WecanuseasimpleimplementaJonofcyclechecking (muljplepathchecking)bbbjustrejectallsearchnodes visijngastatealreadyvisitedbyapreviouslyexpanded node.byproperty(5)weneedkeeponlythefirstpathto astate,rejecjngallsubsequentpaths. Admissibility"without"monotonicity" When" h "is"admissible"but"not"monotonic." TimeandSpacecomplexityremainthesame.Completenessholds. OpJmalitysJllholds(withoutcyclechecking),butneedadifferent argument:dontknowthatpathsareexploredinorderofcost. 101" 102" Admissibility"without"monotonicity" Space"Problems"with"A*" What"about"Cycle"Checking?" NolongerguaranteedwehavefoundanopJmalpathtoanodethe$first$ (mewevisitit. So,cyclecheckingmightnotpreserveopJmality. Tofixthis:forpreviouslyvisitednodes,mustremembercostof previouspath.ifnewpathischeapermustexploreagain. A*hasthesamepotenJalspaceproblemsasBFSorUCS IDA*BIteraJveDeepeningA*issimilartoIteraJve DeepeningSearchandsimilarlyaddressesspaceissues. 103" 104"

27 IDA*"W"Itera0ve"Deepening"A*" ObjecJve:reducememoryrequirementsforA* LikeiteraJvedeepening,butnowthe cutoff isthefbvalue(g+h)rather thanthedepth AteachiteraJon,thecutoffvalueisthesmallestfBvalueofanynodethat exceededthecutoffonthepreviousiterajon Construc0ng"Heuris0cs" Avoidsoverheadassociatedwithkeepingasortedqueueofnodes,andthe openlistoccupiesonlylinearspace. Twonewparameters: curbound(anynodewithabiggerfbvalueisdiscarded) smallestnotexplored(thesmallestfbvaluefordiscardednodesina round)whenopenbecomesempty,thesearchstartsanewroundwith thisbound. EasiertoexpandallnodeswithfBvalueEQUALtothefBlimit.Thisway wecancompute smallestnotexplored moreeasily. 105" 106" Building"Heuris0cs:"Relaxed"Problem" Building"Heuris0cs:"Relaxed"Problem" Oneusefultechniqueistoconsideraneasierproblem,andlet h(n)bethecostofreachingthegoalintheeasierproblem. 8BPuzzlemoves. CanmoveaJlefromsquareAtoBif Aisadjacent(leU,right,above,below)toB andbisblank CanrelaxsomeofthesecondiJons 1. canmovefromatobifaisadjacenttob(ignorewhetherornot posijonisblank) 2. canmovefromatobifbisblank(ignoreadjacency) 3. canmovefromatob(ignorebothcondijons). #3" can#move#from#a#to#b#(ignore#both#condicons).## "leads"to"the"misplaced"0les"heuris0c." Tosolvethepuzzle,weneedtomoveeachJleintoitsfinalposiJon. Numberofmoves=numberofmisplacedJles. Clearlyh(n)=numberofmisplacedJles theh*(n)thecostofanopjmal sequenceofmovesfromn. #1" can#move#from#a#to#b#if#a#is#adjacent#to#b#(ignore#whether#or#not# posicon#is#blank) # "leads"to"the"manhaoan"distance"heuris0c. TosolvethepuzzleweneedtoslideeachJleintoitsfinalposiJon. WecanmoveverJcallyorhorizontally. Numberofmoves=sumoveralloftheJlesofthenumberofverJcaland horizontalslidesweneedtomovethatjleintoplace. Againh(n)=sumofthemanhaandistances h*(n) inarealsolujonweneedtomoveeachjleatleastthatfarandwe canonlymoveonejleatajme. 107" 108"

28 Building"Heuris0cs:"Relaxed"Problem" Building"Heuris0cs:"Relaxed"Problem" ComparisonofIDSandA*(averagetotalnodesexpanded): Depth IDS A*(Misplaced)h1" A*(Manhaan)h2" 10 47, ,473, BBB 39,135 1,641 Leth1=Misplaced,h2=Manhaan Doesh2alwaysexpandfewernodesthanh1? Yes!Notethath2"dominates"h1,i.e.foralln:h1(n) h2(n).from thisyoucanproveh2isfasterthanh1(oncebothareadmissible). Therefore,amongseveraladmissibleheurisJctheonewithhighest valueisthefastest. Theop0malcosttonodesintherelaxedproblemisan admissibleheurisjcfortheoriginalproblem! Proof"Idea:theopJmalsoluJonintheoriginalproblemisa solujonforrelaxedproblem,thereforeitmustbeatleastas expensiveastheopjmalsolujonintherelaxedproblem. SoadmissibleheurisJcscansomeJmesbeconstructedby findingarelaxajonwhoseopjmalsolujoncanbeeasily computed. 109" 110" Building"Heuris0cs:"Paoern"databases" TrytogenerateadmissibleheurisJcsbysolvinga subproblemandstoringtheexactsolujoncostforthat subproblem SeeChapter3.6.3ifyouareinterested. 111"