高级算法优质PPT.ppt

高级算法优质PPT.ppt

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BinarysearchFindanelementinasortedarray:

Divide:

Findandcheckthemiddleelement.Conquer:

Recursivelysearch1subarray.Combine:

Trivial.Example:

Find9BinarysearchFindanelementinasortedarray:

Find9ComplexityforbinarysearchT（n）=1T（n/2）+

（1）#subproblemssubproblemsizecostofdividingandcombiningRecallourMasterMethodSolverecurrencesoftheform:

T（n）=aT（n/b）+f（n）,wherea1,b1,andf（n）isasymptoticallypositive.RecurrenceforbinarysearchT（n）=1T（n/2）+

（1）b=2,a=1Case2（k=0）#subproblemssubproblemsizecostofdividingandcombiningExam2:

PoweringaNumberProblem:

Computean,wherenN.Naivealgorithm:

How?

Complexity?

TheSpotCreativity:

Isthistheonlyandthebestalgorithm?

AnySuggestionsonusingdivide-and-conquerstrategy?

（n）Divide-and-conquer:

Complexity:

T（n）=T（n/2）+

（1）a=1,b=2=n0=1case2（k=0）T（n）=（lgn）PoweringaNumberTheBirthdayParadoxHowmanypeoplemusttherebeinaroomiftherearetwoofthemwerebornonthesamedayoftheyear?

Howmanypeoplemusttherebeinaroomifthereisabigchancethattwoofthemwerebornonthesameday?

Suchasprobabilityofmorethan50%?

IndicatorRandomVariableWeknowthattheprobabilityofisbirthdayandjsbirthdaybothfallonthesamedayris1/n,n=365WedefinetheindicatorrandomvariableXijfor1ijk,byIndicatorRandomVariableThuswehaveEXij=Prpersoniandjhavethesamebirthday=1/n.LettingXbetherandomvariablethatcountsthenumberofpairsofindividualshavingthesamebirthdayTheBirthdayParadoxIfwehaveatleastindividualsinaroom,wecanexpecttwotohavethesamebirthday.Forn=365,ifk=28,theexpectednumberofpairswiththesamebirthdayis（2827）/（2365）1.0356.FibonaccinumbersBackgroundofFibonacci0112358132134.GivemetherecursivedefinitionofFibonacciNumbers?

GivemeanapplicationofFibonaccinumbers.ApplicationofFibonacciThereisastaircaseof10steps,andyoucanstepacrossoneortwostepsasyoulikeeachtime.Howmanywayswehavetosteponthestage?

WhatstheratioofthetwoadjacentnumbersintheFibonacciSequence,thatstosay,whatsthevalueofFn-1/Fnwhenntendsto?

ElliottWavePrinciple:

KeytoMarketBehaviorQuestion:

HowcanwecomputeFibonacciasfastaspossibleincomputer?

NavealgorithmofFibonacciTocomputerFibonaccinumberFn=Fn-1+Fn-2Howmuchtimedoesittake?

SubstitutionmethodRecursion-treemethodMastermethodRecallSubstitutionmethodThemostgeneralmethodGuesstheformofthesolution.Verifybyinduction.Solveforconstants.Substitutionmethodisusedtodeterminetheupperorlowerboundsofrecurrence.AnalyzeofFibonacciTocomputerFibonaccinumberFn=Fn-1+Fn-2Howmuchtimedoesittake?

SubstitutionmethodRecursion-treemethodMastermethodSubstitutionmethodGuessO（2n）Guess=T（n）=Infact,T（n）=（1.618）nQuestionsWhythisrecursivealgorithmtakessomuchtime?

BuildingouttherecursiontreeforF（n）,wecanseethattherearelotsofcommonsubtrees.ToomanyduplicationsthatwastetimeImportantinRecursionCompoundInterestRuleCanweimprovethisalgorithm?

Thinkaboutthebottom-upimplantationofthisrecursivealgorithmComputetheFibonaccinumbersinorder.IterativeTocomputerFibonaccinumberF（n）,Howmuchtimedoesittake?

T（n）=（n）Isthatthebestwecando?

AnyideasonhowwecouldcomputeFibonacciofnfasterthanlineartime?

UsingmathematicalpropertiesofFibonaccinumbersroundedtothenearestinteger,whereWhatwecallNaiverecursivesquaring.TimeComplexity:

Doesthisworkwell?

Thismethodisunreliable,sincefloating-pointarithmeticispronetoround-offerrors.RecursiveSquaringAnothermathematicalpropertyofFibonaccinumbersProofofTheoremRecursiveSquaringAnothermathematicalpropertyofFibonaccinumbersWhatsthecostofcomputingFn?

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