ExampleThe8puzzle ExampleThe8puzzlecontdIstatesintegerlocationsoftilesignoreintermediatepositionsIactionsmoveblankleftrightupdownignoreunjammingetcIgoaltestgoalstategivenIpathcost ID: 451588
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Example:TravelinginRomaniaIOnholidayinRomania;currentlyinAradIFlightleavestomorrowfromBucharestIFormulategoal:beinBucharestIFormulateproblem:states:variouscitiesactions:drivebetweencitiesIFindsolution:sequenceofcities,e.g.,Arad,Sibiu,Fagaras,Bucharest(anysolutionoroptimalsolution?) Example:The8-puzzle Example:The8-puzzle(cont'd)Istates:integerlocationsoftiles(ignoreintermediatepositions)Iactions:moveblankleft,right,up,down(ignoreunjammingetc.)Igoaltest:=goalstate(given)Ipathcost:1permoveINotethattheoptimalsolutionofn-PuzzlefamilyisNP-hard Treesearchalgorithms(cont'd)functionTree-Search(problem,strategy)returnsasolution,orfailureinitializethefrontierusingtheinitialstateofproblemloopdoifthefrontierisemptythenreturnfailurechoosealeafnodeandremoveitfromthefrontierifthenodecontainsagoalstatethenreturnthecorrespondingsolutionexpandthechosennodeandaddtheresultingnodestothefrontierend EvaluatingsearchstrategiesIAstrategyisdenedbypickingtheorderofnodeexpansionIStrategiesareevaluatedalongthefollowingdimensions:Icompleteness|doesitalwaysndasolutionifoneexists?Itimecomplexity|numberofnodesgenerated/expandedIspacecomplexity|maximumnumberofnodesinmemoryIoptimality|doesitalwaysndaleast-costsolution?ITimeandspacecomplexityaremeasuredintermsofIb|maximumbranchingfactorofthesearchtreeId|depthoftheleast-costsolutionIm|maximumdepthofthestatespace(maybe1) UninformedsearchstrategiesUninformedstrategiesuseonlytheinformationavailableintheproblemdenitionIBreadth-rstsearchIUniform-costsearchIDepth-rstsearchIDepth-limitedsearchIIterativedeepeningsearchIBidirectionalsearch Breadth-rstsearchIExpandtheshallowestunexpandednodeIImplementation:frontierisaFIFOqueue,i.e.,newsuccessorsgoatend Progressofbreadth-rstsearchBreadth-rstsearchonasimplebinarytree.Ateachstage,thenodetobeexpandednextisindicatedbyamarker.Thenodesthatarealreadyexploredaregray.Thenodeswithdashedlinesarenotgeneratedyet. Progressofbreadth-rstsearchBreadth-rstsearchonasimplebinarytree.Ateachstage,thenodetobeexpandednextisindicatedbyamarker.Thenodesthatarealreadyexploredaregray.Thenodeswithdashedlinesarenotgeneratedyet. Progressofbreadth-rstsearchBreadth-rstsearchonasimplebinarytree.Ateachstage,thenodetobeexpandednextisindicatedbyamarker.Thenodesthatarealreadyexploredaregray.Thenodeswithdashedlinesarenotgeneratedyet. Progressofbreadth-rstsearchBreadth-rstsearchonasimplebinarytree.Ateachstage,thenodetobeexpandednextisindicatedbyamarker.Thenodesthatarealreadyexploredaregray.Thenodeswithdashedlinesarenotgeneratedyet. Propertiesofbreadth-rstsearchIComplete:Yes(ifbisnite)ITime:b+b2+b3+:::+bd+b(bd�1)=O(bd+1),i.e.,numberofnodesgeneratedisexponentialindISpace:O(bd+1)(keepseverynodeinmemory)IOptimal:Yes(ifcost=1perstep)Spaceisthebigproblem;caneasilygeneratenodesat100MB/secso24hrs=8604GB. Uniform-costsearchIExpandtheleast-costunexpandednodeIImplementation:frontierisaqueueorderedbypathcostIEquivalenttobreadth-rstifstepcostsareallequal Propertiesofuniform-costsearchIComplete:Yes,ifstepcostITime:#ofnodeswithgcostofoptimalsolution,O(b1+bC=c)whereCisthecostoftheoptimalsolutionISpace:#ofnodeswithgcostofoptimalsolution,O(b1+bC=c)IOptimal:Yes|nodesexpandedinincreasingorderofg(n) Depth-rstsearchIExpanddeepestunexpandednodeIImplementation:frontierisaLIFOqueue,i.e.,putsuccessorsatfront Progressofdepth-rstsearch Progressofdepth-rstsearch Progressofdepth-rstsearch Progressofdepth-rstsearch Progressofdepth-rstsearch Progressofdepth-rstsearch Progressofdepth-rstsearch Progressofdepth-rstsearch Progressofdepth-rstsearch Progressofdepth-rstsearch Progressofdepth-rstsearch Progressofdepth-rstsearch Propertiesofdepth-rstsearchIComplete:No:failsininnite-depthspaces,spaceswithloopsModifytoavoidrepeatedstatesalongpath)completeinnitespacesITime:O(bm):terribleifmismuchlargerthandbutifsolutionsaredense,maybemuchfasterthanbreadth-rstISpace:O(bm),i.e.,linearspace!IOptimal:No Depth-limitedsearchfunctionDepth-Limited-Search(problem,limit)returnsasolution,orfailure/cuto returnRecursive-DLS(Make-Node(problem.Initial-State),problem,limit)functionRecursive-DLS(node,problem,limit)returnsasolution,orfailure/cuto ifproblem.Goal-Test(node.State)thenreturnSolution(node)elseiflimit=0thenreturncuto elsecuto -occurred? falseforeachactioninproblem.Actions(node.State)dochild Child-Node(problem,node,action)result Recursive-DLS(child,problem,limit-1)ifresult=cuto thencuto -occurred? trueelseifresult6=failurethenreturnresultifcuto -occurred?thenreturncuto elsereturnfailure IterativedeepeningsearchfunctionIterative-Deepening-Search(problem)returnsasolution,orfailurefordepth 0to1doresult Depth-Limited-Search(problem,depth)ifresult6=cuto thenreturnresult Summaryofalgorithms Criterion Breadth-Uniform-Depth-Depth-Iter. FirstCostFirstLimitedDeep. Complete? YesYesNoYesYes Time O(bd+1)O(b1+bC=c)O(bm)O(bl)O(bd) Space O(bd+1)O(b1+bC=c)O(bm)O(bl)O(bd) Optimal? YesYesNoNoYes SummaryIProblemformulationusuallyrequiresabstractingawayreal-worlddetailstodeneastatespacethatcanfeasiblybeexplored.IThereareavarietyofuninformedsearchstrategiesavailable.IIterativedeepeningsearchusesonlylinearspaceandnotmuchmoretimethanotheruninformedalgorithms.