02-Analysis of Insertion Sort - cs.albany.edupetko/classes/FALL16-CSI403...Insertion sort growth...

12
403: Algorithms and Data Structures Analysis of Insertion Sort Fall 2016 UAlbany Computer Science

Transcript of 02-Analysis of Insertion Sort - cs.albany.edupetko/classes/FALL16-CSI403...Insertion sort growth...

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403:AlgorithmsandDataStructures

AnalysisofInsertionSortFall2016UAlbany

ComputerScience

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Tune-inexercise

• Whatisalgorithmanalysis?–Meanstopredicttheresourcesneededforanalgorithmexecution.

• Whatresourcesareweconcernedwith?– Runningtimeandmemory

• Whydoweneedsuchresourceprediction?– Tobeabletocomparealgorithms– Tobeabletoprovisionresourcesforalgorithmexecution

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Exampleofinsertionsortonaninstance

• Whichisthealgorithm?• Whichistheinput?• Whichistheoutput?• Whatistheinstance?

Thealgorithm

Theinput

Theoutput

<5,2,4,6,1,3>

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Exampleofinsertionsortonaninstance

Takeaminutetothinkonyourownofwhatishappeningateachstep.

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Insertionsort(pseudocode)

Thisstepcanbereachedwheni=0orifA[i]≤key.Inbothcaseskeyisplaceds.t. A[1…i]issorted

Inputarrayis1-based

j indexesthewhole

array

iindexesthesortedsequence

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Insertionsort-- analysis

• Recallthateachprimitiveoperationtakesconstanttime

• Assumetherearen numbersintheinput

• c1,c2,andc3 areconstantsanddonotdependonn

c1

c2c3

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Insertionsort-- analysis

• Assumetherearen numbersintheinput

c2

c1

c3

Whatisthetimeneededforthealgorithmexecution?

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Insertionsort-- analysis• Assumetherearen numbersintheinput

• While loopisexecutedatmostj-1 timesforagiven j,sotimespentinloopisatmost(j-1)c2

• AnyiterationoftheouterFor looptakesatmostc1 +(j-1)c2+c3

• Theoverallrunningtimeofinsertionsortis∑ c1+(j-1)c2+c3𝒏𝒋$𝟐 =d1n2 +d2n+d3

c2

c1

c3

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Wasouranalysistoopessimistic?

• Wejustperformedaworst-caseanalysisofinsertionsort,whichgaveusanupperboundoftherunningtime.

• Wasouranalysistoopessimistic?Inotherwords,arethereinstancesthatwillcausethealgorithmtorunwithquadratictimeinn?– Theworst-caseinstanceisareverse-sortedsequencea1,a2,…,an suchthata1>a2>…>an

• Sinceworst-casesequenceexists,wesaythatouranalysisis“tight”and”notpessimistic”.

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Insertionsortgrowthrate

• Considerinsertionsort’srunningtimeasthefunctiond1n2 +d2n+d3– Thedominantpartofthisfunctionisn2 (i.e.asnbecomeslarge,n2 growsmuchfasterthann)

– Thus,thegrowthrateoftherunningtimeisdeterminedbythen2 term

–WeexpressthisasO(n2)(a.k.a.“big-oh”notation*)–Wecomparealgorithmsintermsoftheirrunningtime

*Tobeformallydefinedlater

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Algorithmcomparison• Whichalgorithmisbetter?

– Weanswerthisquestionbycomparingalgorithms’O()runningtimes.

• Example:ComparealgorithmAandB.Whichoneisbetter?– AlgorithmA:O(n2)– AlgorithmB:O(nlog2(n))

ByCmglee - Ownwork,CCBY-SA4.0,https://commons.wikimedia.org/w/index.php?curid=50321072

• Bismoreefficient.• Intuitivelyn2 growsfaster• Wemightbewrongforsmallinstancesbutwhen nislargeBwillbefaster

• Largesizescomeaboutveryoften(Facebookhas100sofmillionsofusers)

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Announcements

• ReadthroughChapters1and2inthebook• Homework1posted,DueonSep7


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