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CS161:DesignandAnalysisof

Algorithms

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DynamicProgrammingI:

WeightedIntervalScheduling Example:FibonacciNumbers RecurrenceTrees DynamicProgrammingDags WeightedIntervalScheduling

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Example:FibonacciNumbers

F(n)={0ifn=01ifn=1F(n-1)+F(n-2)ifn>1}

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RecursiveAlgorithm

Fib1(n)={Ifn

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RecursiveAlgorithm

RunningTime?Claim:Forn>0,numberofaddionsisF(n)-1Trueforn=1,2

Inducvelyassumetruefork

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RecursiveAlgorithm

RunningTime?(F(n))HowfastdoesF(n)grow?

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FibonacciGrowthRate

Letandbesoluonstox2=x+11.62,thegoldenrao-0.62

Claim:F(n)=(n) StrongerClaim:F(n)=(nn)/()Proof:Trueforn=0,1Assumetruefork

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FibonacciGrowthRate

F(n)=(nn)/() Proof:Trueforn=0,1

Assumetruefork

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RecursiveAlgorithm

RunningTime?(F(n))(1.62n)Growsextremelyrapidly

Example:Mycomputer Runningme1.7x(1.62)nnanoseconds n=30:4milliseconds n=40:0.42seconds n=50:51seconds n=60:1.8hours(projecon) n=70:9.1days(projecon) n=130:93billionyears(projecon)

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Problem

TocomputeF(n):ComputeF(n-1)andF(n-2)CompungF(n-1)requirescompungF(n-2)andF(n-3)

CompungF(n-2)requirescompungF(n-3)andF(n-4)

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Problem

TocomputeF(n):CallF(n-k)atotalofF(k-1)mesWaytoomuchrepeatedwork

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Soluon:Memoizaon

RememberanswerstoF(k)forfuturecalls Keeptrackof(k,F(k))mappings

HashtableArray

Fib2(n)={If(n

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AlternaveApproach

Wearegoingfromtopdown Howaboutgoingboomup:

KeeparrayA,whereA[k]=F(k)Iteravelybuildarray

First,setA[0]=0,A[1]=1 Then,fork=2,,n,setA[k]=A[k-1]+A[k-2]

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IteraveAlgorithm

Fib3(n)={ConstructarrayAoflengthn+1SetA[0]=0,A[1]=1

Fork=2,,n A[k]=A[k-1]+A[k-2]

ReturnA[n]

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IteraveAlgorithm

RunningTime:n-1iteraonsEachstephasa1addion(pretendalladdionsareconstantme)

O(n)meMuchmoretractablenow

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RecursionTree

F(n)

F(n-1) F(n-2)

F(n-3) F(n-3) F(n-4)F(n-2)

F(n-3) F(n-4) F(n-4) F(n-5) F(n-4) F(n-5) F(n-5) F(n-6)

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MergeIdencalNodes

F(n)

F(n-1)

F(n-2)

F(n-3)

F(n-4)

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ProcessBoomUp

F(n)

F(n-1)

F(n-2)

F(n-3)

F(n-4)

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DynamicProgramming

SubproblemshavedagstructureEdgerepresentsprerequisite

SolvesubproblemsintopologicalorderWheneverwesolveasubproblem,wehavealreadysolvedalloftheothersubproblemsweneed

Verygeneral,flexibletool

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WeightedIntervalScheduling

Givensetofnintervals(s(i),f(i)),eachwithaweightw(i)

Goal:picksetofoverlappingintervalswithlargestpossibleweight Ifw(i)=1,wehavetheunweightedschedulingproblem,whichcanbesolvedbygreedy

Doesgreedyworkhere?

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Recallgreedy:pickintervalthatendsearliest

Thisgreedyapproachdoesnotwork2

1

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SaywehaveopmalscheduleS Twopossibilies:

Intervaln(thelastone)isinSIntervalnisnotinS

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SupposeintervalnisnotinSThenSisactuallyopmalforfirstn-1intervalsOtherwise,anyopmalsoluonforfirstn-1intervalsissoluonfornintervalswithhigher

weight

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SupposeintervalnisinSThennointervalthatoverlapsncanbeinSS{n}mustbeopmaloverintervalsthatdontoverlapintervaln

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Suggestthefollowingapproach:SubproblemswillconsistofsubsetsofintervalsIfsubsethassingleinterval,theopmalsoluonforthatsubproblemisjustthatinterval

Otherwise,letTbesomesubset,andlettbethelastinterval

TheopmalforasubsetTiseither: TheopmalforT{t} (TheopmalforT{sintersecngt})+{t}

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Problem

Thereare2nsubsets,sosolvingforallsubsetswilltakeexponenalme

Instead,wewillbeclever:Orderintervalsbyfinishme(i.e.ifi

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Opmalon{1,2,...,k}:Ifintervalkisnotinopmal,thenopmalisjustopmalon{1,2,,k-1}

Ifkisinopmal,thenopmalistheopmalon{1,2,,p(k)},plustheintervalk

Onlyneedtochecktwocases

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WeightedIntervalSchedule:CreatesoluonarraySoflengthn+1CreateweightarrayWoflengthn+1Sortintervalsbyf(i)S[0]={},W[0]=0,S[1]={1},W[1]=w(1)Fork=2,,n:

IfW[k-1]

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RunningTime

Saywehavep(i)values,andlistalreadysorted.

Then,eachiteraontakesonlyO(1),soO(n)overall

Compungp(j)?Obviousalgorithm:O(n2)

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UnderlyingDag

Nodesrepresentsets{1,,k} Pointerfromset{1,,k-1}to{1,,k}forallk Pointerformset{1,,p(k)}to{1,,k}forallk

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DynamicProgrammingOutline

Findgoodsubproblems ExpresssoluontosuproblemkintermsofsoluonstoothersubproblemsSoluontosubproblemkneedstobeefficientlycomputablegivensoluonstoothersubproblems

Solvesubproblemsintopologicalorder

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MoreonFibonacci

Dynamicsoluonisntnecessarilybest OncewevecomputedF(k-1)andF(k-2),wenolongerneedF(k-3),F(k-4),,F(0)

Savespace:onlykeeparoundlasttwocomputedvalues

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MoreonFibonacci

KeeparoundF(k)andF(k-1) Toupdate:

F(k) = F(k1)+F(k 2)

F(k1) = F(k1)

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MoreonFibonacci

KeeparoundF(k)andF(k-1) Toupdate:

F(k)

F(k1)

"

#

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%

&

''=

1 1

1 0

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F(k1)

F(k 2)

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''

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MoreonFibonacci

CanwedobeerthanO(n)addions?F

(n

)F(n1)

"

#$$

%

&'' = 1 11 0

"

#$

%

&'

n1F

(1)F(0)

"

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n1

10

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HowtoComputePowers

SaywehaveasetXwherewecanmulplyelementstogether

Howdowecomputex

n

=x*x*x**x?Obvioussoluon:computexn-1recursively,andmulplybyx

Requiresn-1mulplicaonsintheset

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HowtoComputePowers

Whatifwearecompungx4?Firstcomputey=x2,thencomputey2Only2mulplicaons

Whataboutx8?Computey=x4asabove,thencomputey2Only3mulplicaons

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HowtoComputePowers

Ingeneral,cancomputex2nusingnmulplicaons

Whataboutexponentsthatarenotpowersof2?

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HowtoComputePowers

Pow(x,n)={Ifn=1,returnxIfniseven,returnPow(x*x,n/2)Ifnisodd,returnx*Pow(x*x,(n-1)/2)}

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HowtoComputePowers

Numberofmulplicaons?Atmost2percalltoPowExponentisatleastdividedby2O(logn)mulplicaons

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MoreonFibonacci

CanwedobeerthanO(n)addions?

CancomputeusingO(logn)2x2matrixmulplicaons O(logn)addionsandmulplicaons

F(n

)F(n1)

"

#$$

%

&'' = 1 11 0

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#$

%

&'

n1F

(1)F(0)

"

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%

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n1

10

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TheCatch

Theintegersweareaddingandmulplyingarelarge(exponenal,infact)

Numberofdigits:O(n)

EventhoughO(logn)addionsandmulplicaons,eachaddionand

mulplicaontakesmeuptoO(M(n)),

whereM(n)isthemetomulply2n-digitintegers

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ActualRunningTime

T(n)=T(n/2)+O(M(n)) M(n)isatleastn,sorunningmedominatedbyO(M(n))term

Therefore,T(n)=O(M(n))

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