CSCI360
Introduc/ontoAr/ficialIntelligence
Week2:ProblemSolvingandOp/miza/on
Instructor:Wei-MinShen
StatusCheckandReview
•  Statuscheck
–  HaveyouregisteredinPiazza?
–  HaveyouruntheProject-1?
–  HaveyousubmiMedyourpre-readingreports?
•  Herearesomegoodonesforyourreference
•  Reviewoflastlecture
–  Agent?Typesofagents?
–  Environment?Typesofenvironment?
•  Today’slecture=?
ClassOutline
Wk2Ch3:SolvingProblems
•  Howtorepresentaproblem?
–  states,ac/ons,ini/als,goals,(e.g.,/c-tac-toe)
•  Howtosolveaproblem?
–  Search:fromheretothere,ini/alstogoals
–  Depth-first,breadth-first
•  Howgoodisyoursolu/on?(fig6.1,ALFE)
–  Howgoodisyourstate?Howcostlyisanac/on?
–  Best-first,DynamicProgramming,A*,etc.
–  Canyouguaranteeanything?(op/malvsheuris/c)
•  Howmuchdoyouwanttopayforyoursolu/on?
–  Howdeep/widecanyougo?
•  Predeterminedvsdynamic(e.g.,itera/vedeepening)
–  Onewayormanyways(bi-direc/onal)?
•  Howbigisaproblem?Canyouputthewholeworldinyourhead?
–  TowerofHanoi,chess,robot-and-world,
•  HM:statespaceforTOH,assignvaluesforstateandac/ons
ClassOutline
Wk2Ch4:Op/miza/on
•  Howtofindtheextremesoff(x)?
•  Whyisitsoimportant?
•  Whyisitsohard?(noonehasofferedageneralsolu/on!)
– 
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– 
– 
– 
– 
Whichwaytogo?
Howmuchcanyousee?(localvsglobal)
Howmanypointscanyouremember?
Howbigisyourstep?(skipTHEpoint?)
Howwellcanyouguess?
Howmuchdoyouknowaboutthefunc/on?
Howdoyouknowyouaredone?
Willthefunc/onchangebyitself?
•  HW:yourownanswersfortheaboveques/ons
ThisWeek’sLecture
•  ProblemSolvingandSearchTechniques
•  Op/miza/onTechniques
•  HomeWork1assignments
SolvingProblemsbySearch
•  Howtorepresentareal-worldproblem?
–  Examples,statespace,states,ac/ons,ini/als,goals.
•  Howtosolveaproblem?
–  Search:fromheretothere,ini/alstogoals
–  Depth-first,breadth-first
•  Howgoodisyoursolu/on?(fig6.1,ALFE)
–  Howgoodisyourstate?Howcostlyisanac/on?
–  Best-first,DynamicProgramming,A*,etc.
–  Canyouguaranteeanything?(op/malvsheuris/c)
•  Howmuchdoyouwanttopayforyoursolu/on?
–  Howdeep/widecanyougo?
•  Predeterminedvsdynamic(e.g.,itera/vedeepening)
–  Onewayormanyways(bi-direc/onal)?
•  Howbigisaproblem?Canyouputthewholeworldinyourhead?
–  TowerofHanoi,chess,robot-and-world,
Whatisa“Problem”?
•  Anyideas?
•  Hint:ThinkitasrelatedtoAgent
Whatisa“Problem”?
•  Anyideas?
•  Hint:ThinkitasrelatedtoAgent
–  Percepts,ac/ons,environment,goals,u/li/es
•  Problem:
–  States,ini/alstate,goalstates,
–  Solu/on:asequenceofac/onsleadstothegoal
ProblemExamples:8Puzzle,PAC-Man
Ini/alState
GoalState(s)
TowerofHanoi,ACleaningRobot
Can“dirt”beagoal?
10
TravelExampleandProject-1
StateSpace
Howtorepresentareal-worldproblem
•  GraphSearch
– 
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– 
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– 
Graph
Ver/ces
Edges
Startver/ces
Goalver/ces
Solu/onisa
(minimumcost)
pathfromthe
starttothegoal
—  SearchProblem
— 
— 
— 
— 
— 
— 
Statespace
States
Ac/ons=operators
Ini/alstate
Goalstatesorgoaltest
Solu/onisa(minimum
cost)ac/onsequence
fromthestarttothe
goalstate(orastate
thatsa/sfiesthegoal
test)
ABIGQues/on:
Howdothe“sensors”
and“percepts”enter
thepicture?
VacuumRobotandItsWorld
TravelingfromAradToBucharest
Whatisa“state”here?
StateGraphsvs.SearchTrees
EachNODEinthesearchtreeisan
en/rePATHinthestategraph(note
howmanynodesinthegraphappear
mul/ple/mesinthesearchtree)
G
a
b
c
f
d
s
s
e
p
h
r
q
Wealmostalways
constructbothondemand
–andweconstructasliBle
aspossible
d
b
c
a
a
e
e
h
h
p
q
p
q
c
a
r
p
f
q
G
r
q
q
f
c
a
G
StateSpaceandSearchTree
ASearchTree
Statespace
withac/ons,
costs/rewards
(Fig6.1,ALFE)
•  Anodeinthesearchtreeis
adatastructurecontaining
thestateplusotherdata
17
SearchTreeNodes
SolvingProblemsbySearch
• 
Howtorepresentaproblem?
–  Examples(e.g.,8puzzle,TOH,Roomba,Travel),,states,ac/ons,ini/als,goals.
• 
Howtosolveaproblem?
• 
Howgoodisyoursolu/on?(fig6.1,ALFE)
–  Search:fromheretothere,fromtheini/alstothegoals
–  Depth-first,breadth-first
–  Howgoodisyourstate?Howcostlyisanac/on?
–  Best-first,DynamicProgramming,A*,etc.
–  Canyouguaranteeanything?(op/malvsheuris/c)
• 
Howmuchdoyouwanttopayforyoursolu/on?
–  Howdeep/widecanyougo?
•  Predeterminedvsdynamic(e.g.,itera/vedeepening)
–  Onewayormanyways(bi-direc/onal)?
• 
Howbigisaproblem?Canyouputthewholeworldinyourhead?
–  TowerofHanoi,chess,robot-and-world,
• 
HM:statespaceforTOH,assignvaluesforstateandac/ons.
ProblemSolvingMethods(Overview)
•  Breadth-firstsearch
•  Depth-firstsearch
•  Best-firstsearch
–  Goodness=(past+future)
–  Uniformcost(es/matepastandfuture)
–  A*searchalgorithm(es/matethefuture)
–  DynamicProgramming(backfromthefuture)
Treesearchalgorithms
Basic idea: offline, systematic exploration of simulated
state-space by generating successors of explored states
func(onTREE-SEARCH(problem,strategy)
returnsasolu/on,orfailure
ini/alizethesearchtreeusingtheini/alstateproblem
loopdo
iftherearenocandidatesforexpansion,thenreturnfailure
choosealeafnodeforexpansionaccordingtostrategy
ifthenodecontainsagoalstatethen
returnthecorrespondingsolu/on
elseexpandthenodeandaddresul/ngnodestothesearchtree
end
Breath-firststrategy:theleastdepthnode
Depth-firststrategy:thedeepestdepthnode
Implementa/on(TreeSearch)
func(onTREE-SEARCH(problem)returnasolu/onorfailure
fron/er←MAKE-QUEUE(MAKE-NODE(problem.INITIAL-STATE))
loopdo
ifEMPTY?(fron/er)thenreturnfailure
node←CHOOSE-NODE-By-Strategy(fron/er)
ifproblem.GOAL-TESTappliedtonode.STATEsucceeds
thenreturnSOLUTION(node)
fron/er←INSERT-ALL(EXPAND(node,problem),fron/er)
•  Thefron/eristhesetofnodes
thathavebeengeneratedbut
notyetexpanded
Strategy:Breadth-FirstSearch
•  Expandshallowestnodesfirst
–  Leotorightatanydepth
22
Breath-firstSearchexample
StateGraph
Depth-FirstSearch
•  Expanddeepestnodesfirst
–  Leotorightatanydepth
24
Depth-FirstSearch
•  Expanddeepestnodesfirst
–  Leotorightatanydepth
–  Backtrackwhenhit“terminal”
Depth-firstSearchexample
StateGraph
Ques/on:
Willthedepthgrowforever?
Whereisthe“terminal”?
LoopProblems
GraphSearch(Preven/ngLoop)
func(onGRAPH-SEARCH(problem)returnasolu/onorfailure
fron/er←MAKE-QUEUE(MAKE-NODE(problem.INITIAL-STATE))
explored_set←empty
loopdo
ifEMPTY?(fron/er)thenreturnfailure
node←CHOOSE-NODE-By-Strategy(fron/er)
ifproblem.GOAL-TESTappliedtonode.STATEsucceeds
thenreturnSOLUTION(node)
explored_set←INSERT(node,explored_set)
foreachnew_nodeinEXPAND(node,problem)do
ifNOT(MEMBER?(new_node,fron/er))and
NOT(MEMBER?(new_node,explored_set))
thenfron/er←INSERT(new_node,fron/er)
SolvingProblemsbySearch
• 
Howtorepresentaproblem?
–  Examples(e.g.,8puzzle,TOH,Roomba,Travel),,states,ac/ons,ini/als,goals.
• 
Howtosolveaproblem?
• 
Howgoodisyoursolu/on?(fig6.1,ALFE)
–  Search:fromheretothere,fromtheini/alstothegoals
–  Depth-first,breadth-first
–  Howgoodisyourstate?Howcostlyisanac/on?
–  Best-first,DynamicProgramming,A*,etc.
–  Canyouguaranteeanything?(op/malvsheuris/c)
• 
Howmuchdoyouwanttopayforyoursolu/on?
–  Howdeep/widecanyougo?
•  Predeterminedvsdynamic(e.g.,itera/vedeepening)
–  Onewayormanyways(bi-direc/onal)?
• 
Howbigisaproblem?Canyouputthewholeworldinyourhead?
–  TowerofHanoi,chess,robot-and-world,
• 
HM:statespaceforTOH,assignvaluesforstateandac/ons.
Best-Firstsearchalgorithms
func(onTREE-SEARCH(problem,strategy)
returnsasolu/on,orfailure
ini/alizethesearchtreeusingtheini/alstateproblem
loopdo
iftherearenocandidatesforexpansion,thenreturnfailure
choosealeafnodeforexpansionaccordingtostrategy
ifthenodecontainsagoalstatethen
returnthecorrespondingsolu/on
elseexpandthenodeandaddresul/ngnodestothesearchtree
end
Best-firststrategy:choosethe“best”node:goodness=(pastcost+futurecost)
Whendoweusethisstrategy?
TravelingfromAradToBucharest
Wehavetheknowledgeabouttheactualdrivingdistances
Best-firstSearchexample
Choosethebestnodetoexpand:(pastcost)+(thebestfuturecost)
Sibiu:(140)+(80+97+101)=?
Timisoanra:(118)+(111+70+75+120+138)=?
Zerind:(75)+(71+151+80+97+101)=?
TwoBigQues/onsabout“Best”?
•  Whatifthebestfuturecostisunknowntous?
•  Whatifthebestfuturecostistooexpensivetocompute?
Whatifwedon’tknowthefuture?
•  Goodness=pastcost+thebestfuturecost
–  Whatifthebestfuturecostisunknowntous?
•  Es/mateGoodness:uniformcost(naïve)
•  Es/mateGoodness=pastcost+es/matedfuturecost
–  Evalua/onfunc/onf(n)=g(n)+h(n)
•  g(n)thecost(sofar)toreachthecurrentnoden
•  h(n)es/matedcosttogetfromthecurrentnodentothegoal
•  f(n)es/matedtotalcostofpaththroughntothegoal
TravelingfromAradToBucharest
Supposethefuturedrivingdistancestothegoalisunknowntous
Butweknowthestraight-linedistancestothegoal
Sensor:?
Sensor:?
Usingthees/matedfuturecost
Choosethebestnodetoexpand:(pastcost+es/matedfuturecost)
Sibiu:140+253
Timisoanra:118+329
Zerind:75+374
Whatifoures/ma/oniswrong?
Examplesareplenty
“Myfathertoldmeso”
“Ididdrivethatroadbefore”
“IhaveaniPhone”
“Googlemap”
……
How“wrong”canwetolerate?
Over-es/mate?
Under-es/mate?
Byhowmuch?
Es/mateby“AdmissibleHeuris/cs”
•  Aheuris/cisadmissibleifitneveroveres/matesthe
futurecosttoreachthegoal
•  Admissibleheuris/csare“op/mis/c”
Formally:
1.h(n)≤h*(n)whereh*(n)isthetruecostfromntothegoal
2.h(n)≥0,withh(G)=0foranygoalG
e.g.hSLD(n)(GPS)isadmissiblebecauseitneveroveres/matestheactual
roaddistancetothegoal
38
A*searchalgorithm
func(onTREE-SEARCH(problem,strategy)
returnsasolu/on,orfailure
ini/alizethesearchtreeusingtheini/alstateproblem
loopdo
iftherearenocandidatesforexpansion,thenreturnfailure
choosealeafnodeforexpansionaccordingtostrategy
ifthenodecontainsagoalstatethen
returnthecorrespondingsolu/on
elseexpandthenodeandaddresul/ngnodestothesearchtree
end
Best-firststrategy:choosethebestnode:goodness=(pastcost+futurecost)
A*strategy:choosethe“admissiblyes/matedbest”node:
goodness=(pastcost+admissiblyes/matedfuturecost)
Whatif“futureistoohardtocompute”?
•  Computethebestfuturecostforanode
mustconsiderallpossiblepathsfromthe
nodetothegoal
–  Thisistooexpensivebecausetherearesomany
suchpaths
•  Solu/on
–  UseDynamicProgramming
•  Notcomputepathbypath
•  Butcomputestagebystage(breath-first)
DynamicProgramming(ALFE6.1.1)
Backwardrecursion:computethefuturecostbybackfromthegoal*stagebystage
Backwardrecursionequa/on:
Best-FirstbyDynamicProgramming
func(onTREE-SEARCH(problem,strategy)
returnsasolu/on,orfailure
ini/alizethesearchtreeusingtheini/alstateproblem
loopdo
iftherearenocandidatesforexpansion,thenreturnfailure
choosealeafnodeforexpansionaccordingtostrategy
ifthenodecontainsagoalstatethen
returnthecorrespondingsolu/on
elseexpandthenodeandaddresul/ngnodestothesearchtree
end
Best-firststrategy:choosethebestnode:goodness=(pastcost+futurecost)
A*strategy:choosethebestnode:goodness=(pastcost+admissiblyes/matedfuturecost)
Dynamicprogramming:choosethebestnode:goodness=(pastcost+V(s))
Evalua/onofsearchstrategies
• 
Asearchstrategyisdefinedbypickingtheorderofnodeexpansion.
• 
Searchalgorithmsarecommonlyevaluatedaccordingtothefollowingfourcriteria:
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– 
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– 
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Completeness:doesitalwaysfindasolu/onifoneexists?
Timecomplexity:howlongdoesittakeasfunc/onofnumberofnodes?
Spacecomplexity:howmuchmemorydoesitrequire?
Op(mality:doesitguaranteetheleast-costsolu/on?
Timeandspacecomplexityaremeasuredintermsof:
–  b–maxbranchingfactorofthesearchtree
–  d–depthoftheleast-costsolu/on
–  m–maxdepthofthesearchtree(maybeinfinity)
SolvingProblemsbySearch
• 
Howtorepresentaproblem?
–  Examples(e.g.,8puzzle,TOH,Roomba,Travel),,states,ac/ons,ini/als,goals.
• 
Howtosolveaproblem?
• 
Howgoodisyoursolu/on?(fig6.1,ALFE)
–  Search:fromheretothere,fromtheini/alstothegoals
–  Depth-first,breadth-first
–  Howgoodisyourstate?Howcostlyisanac/on?
–  Best-first,DynamicProgramming,A*,etc.
–  Canyouguaranteeanything?(op/malvsheuris/c)
• 
Howmuchdoyouwanttopayforyoursolu/on?
–  Howdeep/widecanyougo?
•  Predeterminedvsdynamic(e.g.,itera/vedeepening)
–  Onewayormanyways(bi-direc/onal)?
• 
Howbigisaproblem?Canyouputthewholeworldinyourhead?
–  TowerofHanoi,chess,robot-and-world,
• 
HM:statespaceforTOH,assignvaluesforstateandac/ons.
OtherSearch-relatedTechniques
•  Itera/vedeepening
–  Limitthesearchdepth,incrementallyincreaseit
•  Bi-direc/onalsearch
–  Searchfromtheini/altothegoal,aswellas
fromthegoaltotheini/al
SolvingProblemsbySearch
• 
Howtorepresentaproblem?
–  Examples(e.g.,8puzzle,TOH,Roomba,Travel),,states,ac/ons,ini/als,goals.
• 
Howtosolveaproblem?
• 
Howgoodisyoursolu/on?(fig6.1,ALFE)
–  Search:fromheretothere,fromtheini/alstothegoals
–  Depth-first,breadth-first
–  Howgoodisyourstate?Howcostlyisanac/on?
–  Best-first,DynamicProgramming,A*,etc.
–  Canyouguaranteeanything?(op/malvsheuris/c)
• 
Howmuchdoyouwanttopayforyoursolu/on?
–  Howdeep/widecanyougo?
•  Predeterminedvsdynamic(e.g.,itera/vedeepening,fixeddepth)
–  Onewayormanyways(bi-direc/onal)?
• 
Howbigisaproblem?Canyouputthewholeworldinyourhead?
–  TowerofHanoi,chess,robot-and-world,
• 
HM:statespaceforTOH,assignvaluesforstateandac/ons.
Complexity:Howbigisaproblem?
WhenyousolveN=64,
itwouldbetheendoftheworld!
Timecomplexityofdepth-first
•  In the worst case:
•  the (only) goal node may be on the right-most branch,
b0
b1
d
b
G
bd-1
bd
•  Time complexity = bd + bd-1 + … + 1 = bd+1 -1
b-1
•  Thus: O(bd)
CS561-Lectures3-4-Macskassy-Spring
2011
48
Spacecomplexityofdepth-first
•  Largest number of nodes in QUEUE is reached in bottom leftmost node.
•  Example: m = 3, b = 3 :
G
...
•  QUEUE contains all
nodes. Thus: 7.
•  In General: ((b-1) * m) + b
•  Order: O(m*b)
49
Comparinguninformedsearchstrategies
Criterion Time
Space Op/mal?
Complete?
– 
– 
– 
– 
Breadth- Uniform Depth- Depth- Itera/ve Bidirec/onal
first
cost
first
limited deepening(ifapplicable)
b^d
b^d
b^m
b^l
b^d
b^(d/2)
b^d
b^d
bm
bl
bd
b^(d/2)
Yes
Yes
No
No
Yes
Yes
Yes
Yes
No
Yes,
ifl≥d
Yes
Yes
b–maxbranchingfactorofthesearchtree
d–depthoftheleast-costsolu/on
m–maxdepthofthestate-space(maybeinfinity)
l–depthcutoff
50
Summary
•  Problemformula/onusuallyrequiresabstrac/ngaway
real-worlddetailstodefineastatespacethatcanbe
exploredusingcomputeralgorithms.
•  Onceproblemisformulatedinabstractform,complexity
analysishelpsuspickingoutbestalgorithmtosolve
problem.
•  Varietyofuninformedsearchstrategies;differenceliesin
methodusedtopicknodethatwillbefurtherexpanded.
•  Itera/vedeepeningsearchonlyuseslinearspaceandnot
muchmore/methanotheruniformedsearchstrategies.
CS561,Lectures3-5
51
SolvingProblemsbySearch
• 
Howtorepresentaproblem?
–  Examples(e.g.,8puzzle,TOH,Roomba,Travel),,states,ac/ons,ini/als,goals.
• 
Howtosolveaproblem?
• 
Howgoodisyoursolu/on?(fig6.1,ALFE)
–  Search:fromheretothere,fromtheini/alstothegoals
–  Depth-first,breadth-first
–  Howgoodisyourstate?Howcostlyisanac/on?
–  Best-first,DynamicProgramming,A*,etc.
–  Canyouguaranteeanything?(op/malvsheuris/c)
• 
Howmuchdoyouwanttopayforyoursolu/on?
–  Howdeep/widecanyougo?
•  Predeterminedvsdynamic(e.g.,itera/vedeepening)
–  Onewayormanyways(bi-direc/onal)?
• 
Howbigisaproblem?Canyouputthewholeworldinyourhead?
–  TowerofHanoi,chess,robot-and-world,
• 
HM:statespaceforTOH,assignvaluesforstateandac/ons.
HomeWork
•  TowerofHanio(n=3)
–  1.Writethestatespace,states,ac/ons
–  2.Assumethecostofanac/ontomoveadiskis
theweightofthedisk,computethe“bestfuture
cost”V(s0)forthestartstateusingdynamic
programming
Startstates0
goal
Theweightofthedisk:disk1=1,disk2=2,disk3=3