ExamReviewandSolving
HardProblems
ExamReview
• Anyquestions?
• Anytopicsyouwanttoreview?
Solvinghardproblems
• Threekeymethods:
• Backtracking
• Branch-and-bound
• Approximation
Backtracking
• Consideratreeofallchoicesthathavebeenmadesofar
• Specifically,decomposetheproblemintoaseriesofchoiceswhereat
eachstagethereareanumberofoptionstochoosefrom
• Eachnodemaybeeitherpromisingornon-promising
• Promisingnodesmayleadtoasolution
• Non-promisingnodescannotleadto asolution
• Examinepromisingnodes,creatingchildrenuntilasolutionisfound
• OftendoneinaDFSmanner
n-Queens
• Givennqueensandannxnchessboard,positionthequeenssuch
thatnotwoqueenscanattackeachother
n-Queens:keyobservation
• Eachrowwillonlyhaveasinglequeenonit
• Theproblemofplacingnqueensonannxnboardgivesusnchoices:
•
•
•
•
Whatcolumntoplacethequeeninrow1
Whatcolumntoplacethequeeninrow2
...
Whatcolumntoplacethequeeninrown
• Let'slookatanexampleforn=4...
n-Queens:keyobservation
• Eachrowwillonlyhaveasinglequeenonit
• Theproblemofplacingnqueensonannxnboardgivesusnchoices:
•
•
•
•
Whatcolumntoplacethequeeninrow1
Whatcolumntoplacethequeeninrow2
...
Whatcolumntoplacethequeeninrown
• Let'slookatanexampleforn=5...
Hamiltoniancircuit
• Visitallnodesinagraphstartingandendingatthesamevertex,and
visitingnovertexmorethanonce(asidefromthestartingvertex)
• Itdoesn'tmatterwhatvertexwestartat
• Let'sgothroughanexample...
Backtracking
• Backtrackingworkswhenwehaveanumberofstepsandeachstep
wehaveanumberofoptionstochoosebetween
• Workswellforfindinga solution
• Doesn'tworkwellforfindinganoptimalsolution
Branch-and-bound
• Similartobacktracking,butforsituationswherewehaveanobjective
functionwearetryingtooptimize
• Ateachstep,determinealowerbound(inthecaseofminimization)
orupperbound(inthecaseofmaximization)ifachoiceismade
• Ifwe'vealreadyseenasolutionbetterthanthis,noneedtoexamine
thispath
Assignmentproblem
• Givenagroupofpeopleandagroupofjobs,andagridspecifying
howmuchitwouldcostforeachpersontoperformthejob,findthe
bestcombination
Assignmentproblem
• Givenagroupofpeopleandagroupofjobs,andagridspecifying
howmuchitwouldcostforeachpersontoperformthejob,findthe
bestcombination
Job1
Job2
Job3
Job4
Aragorn
9
2
7
8
Beren
6
4
3
7
Círdan
5
8
1
8
Déagol
7
6
9
4
(0-1)Knapsackproblem
• Givenasetofitemswhereeachitemhasaweightandavalue
associatedwithit,pickupblahblahblahyouknowthisalready.
weightlimit=10
Weight
Value
w/v
Diamond 4
$40
10
Beryl
7
$42
6
Agate
5
$25
5
Coal
3
$12
4