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IntroductiontoAlgorithms

6.046J/18.401J

LECTURE1

AnalysisofAlgorithms

•Insertionsort

•Asymptoticanalysis

•Mergesort

•Recurrences

Prof.CharlesE.Leiserson

Courseinformation

1.Staff8.Coursewebsite

2.Distancelearning9.Extrahelp

3.Prerequisites10.Registration

4.Lectures11.Problemsets

5.Recitations12.Describingalgorithms

6.Handouts13.Gradingpolicy

7.Textbook14.Collaborationpolicy

September7,2005IntroductiontoAlgorithmsL1.2

Analysisofalgorithms

Thetheoreticalstudyofcomputer-program

performanceandresourceusage.

Whafsmoreimportantthanperformance?

modularity•user-friendliness

correctness•programmertime

maintainability•simplicity

functionality•extensibility

robustness•reliability

September7,2005IntroductiontoAlgorithmsL1.3

Whystudyalgorithmsand

、、7performance?

•Algorithmshelpustounderstandscalability.

•Performanceoftendrawsthelinebetweenwhat

isfeasibleandwhatisimpossible.

•Algorithmicmathematicsprovidesalanguage

fbrtalkingaboutprogrambehavior.

•Performanceisthecurrencyofcomputing.

•Thelessonsofprogramperformancegeneralize

toothercomputingresources.

•Speedisfun!

September7,2005IntroductiontoAlgorithmsLI.4

ALGORITHMS

Theproblemofsorting

Input:sequence〈%,%夕・・・，％〉°fnumbers.

Output:permutation*，…，such

thata\<a\<--<an.

Example:

Input:824936

Output:234689

September7,2005IntroductiontoAlgorithmsL1.5

ALGORITHMS

.0mmmD

Insertionsort

rINSERTION-SORT(A,n)>A[1..n]

for/<—2ton

dokey^A[j]

“pseudocode"\ijj—1

whilei>0andA[i]>key

doA[i+1]A[i]

i«-z—1

A\i+\}=key

September7,2005IntroductiontoAlgorithmsL1.6

ALGORITHMS

.0mmmD

Insertionsort

rINSERTION-SORT(A,n)>A[1..n]

for/<—2ton

dokey^A[j]

“pseudocode"\ijj—1

whilei>0andA[i]>key

doA[i+1]A[i]

i«-z—1

A\i+\}=key

September7,2005IntroductiontoAlgorithmsL1.7

ALGORITHMS

V_____

：、、，Exampleofinsertionsort

824936

September7,2005IntroductiontoAlgorithmsL1.8

ALGORITHMS

V_____

：、、，Exampleofinsertionsort

824936

September7,2005IntroductiontoAlgorithmsL1.9

ALGORITHMS

V_____

：、、，Exampleofinsertionsort

824936

284936

September7,2005IntroductiontoAlgorithmsLI.10

ALGORITHMS

：、、♦Exampleofinsertionsort

824936

284936

September7,2005IntroductiontoAlgorithmsLI.11

ALGORITHMS

V_____

Exampleofinsertionsort

September7,2005IntroductiontoAlgorithmsLI.12

Exampleofinsertionsort

September7,2005IntroductiontoAlgorithmsLI.13

Exampleofinsertionsort

September7,2005IntroductiontoAlgorithmsLI.14

Exampleofinsertionsort

September7,2005IntroductiontoAlgorithmsLI.15

ALGORITHMS

Exampleofinsertionsort

824936

284936

*—

248936

September7,2005IntroductiontoAlgorithmsLI.16

ALGORITHMS

Exampleofinsertionsort

824936

284936

*—

248936

September7,2005IntroductiontoAlgorithmsLI.17

ALGORITHMS

V_____

：、、，Exampleofinsertionsort

824936

284936

248936

234689done

September7,2005IntroductiontoAlgorithmsLI.18

ALGORITHMS

：、「Runningtime

•Therunningtimedependsontheinput:an

alreadysortedsequenceiseasiertosort.

•Parameterizetherunningtimebythesizeof

theinput,sinceshortsequencesareeasierto

sortthanlongones.

•Generally,weseekupperboundsonthe

runningtime,becauseeverybodylikesa

guarantee.

September7,2005IntroductiontoAlgorithmsLI.19

ALGORITHMS

.—

：、、，Kindsofanalyses

Worst-case:(usually)

•T(n)=maximumtimeofalgorithm

onanyinputofsizen.

Average-case:(sometimes)

•T(n)=expectedtimeofalgorithm

overallinputsofsizen.

•Needassumptionofstatistical

distributionofinputs.

Best-case:(bogus)

•Cheatwithaslowalgorithmthat

worksfastonsomeinput.

September7,2005IntroductiontoAlgorithmsLI.20

ALGORITHMS

Machine-independenttime

Whatisinsertionsort'sworst-casetime?

•Itdependsonthespeedofourcomputer:

•relativespeed(onthesamemachine)5

•absolutespeed(ondifferentmachines).

BIGIDEA:

•Ignoremachine-dependentconstants.

•LookatgrowthofT(n)as〃一oc.

^AsymptoticAnalysis^

September7,2005IntroductiontoAlgorithmsLI.21

ALGORITHMS

©-notation

Math:

0(g(n))={/(〃)：thereexistpositiveconstantsc19c2,and

n0suchthat0<Cjg(n)</(n)<c2g(n)

fbralln>n0}

Engineering:

•Droplow-orderterms;ignoreleadingconstants.

•Example:3"+90/—5〃+6046=0(^3)

September7,2005IntroductiontoAlgorithmsLI.22

ALGORITHMS

：、、，Asymptoticperformance

Whenngetslargeenough,a0(zz2)algorithm

alwaysbeatsa0(H3)algorithm.

•Weshouldn'tignore

asymptoticallyslower

algorithms,however.

•Real-worlddesign

situationsoftencallfora

carefulbalancingof

engineeringobjectives.

•Asymptoticanalysisisa

usefultooltohelpto

structureourthinking.

September7,2005IntroductiontoAlgorithmsLI.23

ALGORITHMS

Insertionsortanalysis

Worstcase:Inputreversesorted.

T(〃)=[arithmeticseries]

六2

Averagecase:Allpermutationsequallylikely.

-(〃)=£。(〃2)=®伍2)

j=2

Isinsertionsortafastsortingalgorithm?

•Moderatelyso,forsmalln.

•Notatall,forlargen.

September7,2005IntroductiontoAlgorithmsLI.24

ALGORITHMS

，Mergesort

MERGE-SORTA[1..n\

1.If/z=1,done.

2.Recursivelysort4[1..「〃/2]]

andA[[n/2~\+l..n].

"Merge"the2sortedlists.

Keysubroutine:MERGE

September7,2005IntroductiontoAlgorithmsLI.25

ALGORITHMS

、、、、Mergingtwosortedarrays

2012

1311

September7,2005IntroductiontoAlgorithmsL1.26

ALGORITHMS

、、、、Mergingtwosortedarrays

2012

1311

September7,2005IntroductiontoAlgorithmsLI.27

ALGORITHMS

.annxnxxxD

Mergingtwosortedarrays

2012III2012

13111311

7979

芋|2

September7,2005IntroductiontoAlgorithmsL1.28

ALGORITHMS

Mergingtwosortedarrays

20122012

13111311

7979

2TT

September7,2005IntroductiontoAlgorithmsLI.29

ALGOMTHMS

.annxnnrc

Mergingtwosortedarrays

2012

1311

September7,2005IntroductiontoAlgorithmsL1.30

Mergingtwosortedarrays

201220

131113

127

September7,2005IntroductiontoAlgorithmsL1.31

Mergingtwosortedarrays

20122012

13111311

September7,2005IntroductiontoAlgorithmsL1.32

Mergingtwosortedarrays

20122012

13111311

1279

September7,2005IntroductiontoAlgorithmsL1.33

Mergingtwosortedarrays

201212

September7,2005IntroductiontoAlgorithmsL1.34

Mergingtwosortedarrays

20122012

1311

12711

September7,2005IntroductiontoAlgorithmsL1.35

Mergingtwosortedarrays

201220122012

131113

111

September7,2005IntroductiontoAlgorithmsL1.36

ALGORiTHJvlS

.amxxnxnD

Mergingtwosortedarrays

20122012

1311

11112

September7,2005IntroductiontoAlgorithmsL1.37

ALGORITHMS

.ami™

Mergingtwosortedarrays

1112

Time=0(77)tomergeatotal

ofnelements(lineartime).

September7,2005IntroductiontoAlgorithmsL1.38

ALGORITHMS

Analyzingmergesort

T®MERGE-SORT^4[1..n\

「6⑴1.If扑=19done.

（2T（〃/2）2.Recursivelysort4[1..「〃/21]

Abuse/and/[[n/2~]+l..n].

/。⑺

3.“Merge”the2sortedlists

Sloppiness:ShouldbeT（「〃/2]）+T（5/2」），

butitturnsoutnottomatterasymptotically.

September7,2005IntroductiontoAlgorithmsL1.39

ALGORITHMS

Recurrenceformergesort

r0(i)if/?=i；

T(〃)=

Y、2T(〃/2)+05)if〃>1.

•Weshallusuallyomitstatingthebase

casewhen7\n)=0(1)fbrsufficiently

smalln,butonlywhenithasnoeffecton

theasymptoticsolutiontotherecurrence.

•CLRSandLecture2provideseveralways

tofindagoodupperboundonT(n).

September7,2005IntroductiontoAlgorithmsLI.40

ALGORITHMS

Recursiontree

SolveT(n)=2T(M2)+cn,wherec>0isconstant.

September7,2005IntroductiontoAlgorithmsL1.41

ALGORITHMS

Recursiontree

SolveT(n)=2T(M2)+cn,wherec>0isconstant.

丁⑺

September7,2005IntroductiontoAlgorithmsLI.42

ALGORITHMS

.U一3

二，Recursiontree

SolveT(n)=2T(n/2)+cn,wherec>0isconstant.

T(〃/2)T(〃/2)

September7,2005IntroductiontoAlgorithmsLI.43

ALGORITHMS

Recursiontree

SolveT(n)=2T(M2)+cn,wherec>0isconstant.

cn/2cn/2

//、

T(T?/4)T(M4)T(TI/4)T(〃/4)

September7,2005IntroductiontoAlgorithmsLI.44

ALGORITHMS

Recursiontree

Solver(n)=2T(M2)+cn.wherec>0isconstant.

cn/4cnJ4c〃/4cnl4

©(1)

September7,2005IntroductiontoAlgorithmsLI.45

ALGORITHMS

Recursiontree

Solver(n)=2T(M2)+cn.wherec>0isconstant.

"—1g"cn14cnl4cn/4cn/4

©(1)

September7,2005IntroductiontoAlgorithmsLI.46

ALGORITHMS

Recursiontree

SolveT(n)=2T(〃/2)+cn,wherec>0isconstant.

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