cse 373: data structures and algorithms...the traveling salesperson problem seattle tehran beijing...

Instructor:LiliandeGreefQuarter:Summer2017

CSE373:DataStructuresandAlgorithmsLecture21:FinishSorting,PvsNP

Today

• Announcements• Finishupsorting• RadixSort• Finalcommentsonsorting

• ComplexityTheory:P=?NP

Announcements

• FinalExam:• Nextweek• Duringusuallecturetime(10:50am- 11:50am)• Cumulative(soallmaterialwe’vecoveredinclassisfairgame)• ...butwithemphasisonmaterialcoveredafterthemidterm• Date:Friday

BacktoSortingAlgorithms

TheBigPicture

Surprisingamountofjuicycomputerscience:2-3lectures…Simple

algorithms:O(n2)

Fancieralgorithms:O(n log n)

Comparisonlower bound:W(n log n)

Specializedalgorithms:

O(n)

Handlinghuge data

sets

Insertion sortSelection sortShell sort…

Heap sortMerge sortQuick sort (avg)…

Bucket sortRadix sort

Externalsorting

How???• Changethemodel– assumemorethan“compare(a,b)”

Radixsort

• Radix=“thebaseofanumbersystem”• Exampleswilluse10becauseweareusedtothat• Inimplementationsuselargernumbers

• Forexample,forASCIIstrings,mightuse128

• Idea:• Bucketsortononedigitatatime

• Numberofbuckets=radix• Startingwithleast significantdigit• Keepingsortstable

• Doonepassperdigit• Invariant:Afterk passes(digits),thelastk digitsaresorted

• Aside:Originsgobacktothe1890U.S.census

RadixSort:Example

Input:

4785379

72133814367

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

Output:

Firstpass:bucketsortbyone’sdigit

Secondpass:stablebucketsortbyten’sdigit

Thirdpass:stablebucketsortbyhundred’sdigit

Analysis

Inputsize:nNumberofbuckets=Radix:BNumberofpasses=“Digits”:P

Workperpassis1bucketsort:

Totalworkis

Comparedtocomparisonsorts,sometimesawin,butoftennot• Example:StringsofEnglishlettersuptolength15

• Run-timeproportionalto:15*(52+n)• Thisislessthann lognonlyifn >33,000• Ofcourse,cross-overpointdependsonconstantfactorsoftheimplementations

• Andradixsortcanhavepoorlocalityproperties

CommentsonSortingAlgorithms

Sortingmassivedata

• Needsortingalgorithmsthatminimizedisk/tapeaccesstime:• QuicksortandHeapsortbothjumpalloverthearray,leadingtorandomdiskaccesses• Mergesortscanslinearlythrougharrays,leadingto(relatively)sequentialdiskaccess

• Mergesortisthebasisofmassivesorting

• Mergesortcanleveragemultipledisks

ExternalMergeSort• Sort900MBusing100MBRAM

• Read100MBofdataintomemory• Sortusingconventionalmethod(e.g.quicksort)• Writesorted100MBtotempfile• Repeatuntilalldatainsortedchunks(900/100=9total)

• Readfirst10MBofeachsortedchuck,mergeintoremaining10MB• writingandreadingasnecessary• Singlemergepassinsteadoflogn• Additionalpasshelpfulifdatamuchlargerthanmemory

• Parallelismandbetterhardwarecanimproveperformance• Distributionsorts(similartobucketsort)arealsoused

Wrap-uponSorting• SimpleO(n2)sortscanbefastestforsmalln

• Insertionsort(latterlinearformostly-sorted)• Good“belowacut-off”fordivide-and-conquersorts

• O(nlog n)sorts• Heapsort,in-place,notstable,notparallelizable• Mergesort,notinplacebutstableandworksasexternalsort• Quicksort,inplace,notstableandO(n2)inworst-case

• Oftenfastest,butdependsoncostsofcomparisons/copies

• W (n log n) isworst-caseandaveragelower-boundforsortingbycomparisons

• Non-comparisonsorts• Bucketsortgoodforsmallnumberofpossiblekeyvalues• Radixsortusesfewerbucketsandmorephases

• Bestwaytosort?

ComplexityTheory:PvsNPJustasmalltasteofComplexityTheory

“Easy”ProblemsfortheComputer

Sortingalistofnnumbers

Multiplyingtwonxnmatrices

3 5 2 71 6 8 92 4 6 109 3 2 12

1 5 5 45 12 8 67 6 1 59 23 5 8

=n

n n

n

n

“Easy”ProblemsfortheComputer

ShortestPathAlgorithm MinimumSpanningTreeAlgorithms

A B

CD

F

E

G

2

12 5

11

1

2 65 3

10

A B

DC

F H

E

G

2 2 3

110 23

111

7

1

9

2

4 5

Edsgar Dijkstra

“Hard”ProblemsfortheComputer

TheKnapsackProblem

Iwanttocarryasmuchmoney’sworthasIcanthatstillfitsinmybag!WhatdoIpack?


TheTravelingSalespersonProblem

Seattle

Tehran

Beijing

Berlin

Moscow

SãoPaulo

54116775

6685

52175058

4583

6356

2185

1004

3608

7325 1533

34887577

10933

I’llleaveSeattletosellgoods,visitingeachcityonlyonce,andreturntoSeattle.What’stheshortestroute?


FindaHamiltonianpath(apaththatvisitseachvertexexactlyonce)

(nevermindweightsorevenreturningtoourstartingpoint!)

Comparingn2 vs2n

Thealien’scomputerperforms109 operations/sec

n=10 n=30 n=50 n=70

n2 100<1sec

900<1sec

2500<1sec

4900<sec

2n 1024<1sec

n! 3628800<1sec

1091sec

101511.6days

102131,688years

1016 years 1048 years 1083 years(105 xageoftheuniverse!)

“Easy”vs“Hard”ProblemsfortheComputer

“PolynomialTime”=“Efficient”

Isanalgorithm“efficient”with…

O(n)? O(nlogn)?O(n2)? O(n10)? O(nlog n)? O(2n)? O(n!)?

PolynomialTime?• Soweknowtherearepolynomialtimealgorithmsto

• Sortnumbers• Multiplynxnmatrices• Findtheshortestpathinagraph• Findtheminimumspanningtree• … andmore

• Butthemilliondollarquestionis…aretherepolynomialtimealgorithmstosolve• TheKnapsackProblem?• TheTravelingSalespersonProblem?• FindingHamiltonianPaths?• … andthousandsmore!

P=NP?

AndthereareproblemsevenharderthanNP!

P(PolynomialTime)

sort

search

Dijkstra’s

Prim’s

Kruskal’s

NP(NondeterministicPolynomialTime)

HamiltonianPath

TravelingSalesperson

KnapsacknxnxnSudoku

SAT

RelevanceofP=NP

NPcontainslotsofproblemswedon’tknowtobeinP• ClassroomScheduling• Packingobjectsintobins• Schedulingjobsonmachines• Findingcheaptoursvisitingasubsetofcities• Findinggoodpacketroutingsinnetworks• Decryption...

Withthisknowledge,wecanavoidsaying…

“Ican’tfindanefficientalgorithm.IguessI’mtoodumb.”

Cartooncourtesyof“ComputersandIntractability:AGuidetotheTheoryofNP-Completeness” byM.Garey andD.Johnson

Butknowitisn’twisetosay…

“Ican’tfindanefficientalgorithmbecausenosuchalgorithmispossible!”


And,instead,proveit’sinNPtothensay…

“Ican’tfindanefficientalgorithm,butneithercanallthesefamouspeople.”


PreparingforFinalExam

FinalExamStudyStrategies

PracticeProblem

Given a value ‘x’ and an array of integers, determine whether two of the numbers add up to ‘x’

Questionsyoushouldhaveaskedme:

1) Isthearrayinanyparticularorder?2) ShouldIconsiderthecasewhereaddingtwolargenumberscouldcausean

overflow?3) Isspaceafactor,inotherwords,canIuseanadditionalstructure(s)?4) Isthismethodgoingtobecalledfrequentlywithdifferent/thesamevalueof‘x’?5) AbouthowmanyvaluesshouldIexpecttoseeinthearray,oristhatunspecified?6) Will‘x’alwaysbeapositivevalue?7) CanIassumethearraywon’talwaysbeempty,whatifitsnull?

PracticeProblem

Given a value ‘x’ and an array of integers, determine whether two of the numbers add up to ‘x’

cse 373: data structures and algorithms...the traveling salesperson problem seattle tehran beijing...

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