big-o algorithm complexity cheat sheet

2/19/2015 BigOAlgorithmComplexityCheatSheet

http://bigocheatsheet.com/ 1/12

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KnowThyComplexities!Hithere!ThiswebpagecoversthespaceandtimeBigOcomplexitiesofcommonalgorithmsusedinComputerScience.Whenpreparingfortechnicalinterviewsinthepast,Ifoundmyselfspendinghourscrawlingtheinternetputtingtogetherthebest,average,andworstcasecomplexitiesforsearchandsortingalgorithmssothatIwouldn'tbestumpedwhenaskedaboutthem.Overthelastfewyears,I'veinterviewedatseveralSiliconValleystartups,andalsosomebiggercompanies,likeYahoo,eBay,LinkedIn,andGoogle,andeachtimethatIpreparedforaninterview,Ithoughttomyself"Whyohwhyhasn'tsomeonecreatedaniceBigOcheatsheet?".So,tosaveallofyoufinefolksatonoftime,Iwentaheadandcreatedone.Enjoy!

Good Fair Poor

Searching

Algorithm DataStructure TimeComplexity SpaceComplexityAverage Worst Worst

DepthFirstSearch(DFS) Graphof|V|verticesand|E|edges O(|E|+|V|) O(|V|)

BreadthFirstSearch(BFS) Graphof|V|verticesand|E|edges O(|E|+|V|) O(|V|)

Binarysearch Sortedarrayofnelements O(log(n)) O(log(n)) O(1)Linear(BruteForce) Array O(n) O(n) O(1)ShortestpathbyDijkstra,usingaMinheapaspriorityqueue

Graphwith|V|verticesand|E|edges

O((|V|+|E|)log|V|)

O((|V|+|E|)log|V|)

O(|V|)

ShortestpathbyDijkstra,Graphwith|V|verticesand

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usinganunsortedarrayaspriorityqueue

|E|edges O(|V|^2) O(|V|^2) O(|V|)

ShortestpathbyBellmanFord Graphwith|V|verticesand|E|edges O(|V||E|) O(|V||E|) O(|V|)

Sorting

Algorithm DataStructure TimeComplexity WorstCaseAuxiliarySpaceComplexityBest Average Worst Worst

Quicksort Array O(nlog(n)) O(nlog(n)) O(n^2) O(n)Mergesort Array O(nlog(n)) O(nlog(n)) O(nlog(n)) O(n)Heapsort Array O(nlog(n)) O(nlog(n)) O(nlog(n)) O(1)BubbleSort Array O(n) O(n^2) O(n^2) O(1)InsertionSort Array O(n) O(n^2) O(n^2) O(1)SelectSort Array O(n^2) O(n^2) O(n^2) O(1)BucketSort Array O(n+k) O(n+k) O(n^2) O(nk)RadixSort Array O(nk) O(nk) O(nk) O(n+k)

DataStructures

DataStructure TimeComplexity SpaceComplexityAverage Worst Worst

Indexing Search Insertion Deletion Indexing Search Insertion DeletionBasicArray O(1) O(n) O(1) O(n) O(n)DynamicArray O(1) O(n) O(n) O(n) O(1) O(n) O(n) O(n) O(n)SinglyLinkedList O(n) O(n) O(1) O(1) O(n) O(n) O(1) O(1) O(n)

DoublyLinkedList O(n) O(n) O(1) O(1) O(n) O(n) O(1) O(1) O(n)

SkipList O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(n) O(n) O(n) O(n) O(nlog(n))

HashTable O(1) O(1) O(1) O(n) O(n) O(n) O(n)BinarySearchTree O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(n) O(n) O(n) O(n) O(n)

CartresianTree O(log(n)) O(log(n)) O(log(n)) O(n) O(n) O(n) O(n)BTree O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(n)RedBlackTree O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(n)

SplayTree O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(n)AVLTree O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(n)



Heaps

Heaps TimeComplexityHeapify FindMax ExtractMax IncreaseKey Insert Delete Merge

LinkedList(sorted) O(1) O(1) O(n) O(n) O(1) O(m+n)LinkedList(unsorted) O(n) O(n) O(1) O(1) O(1) O(1)BinaryHeap O(n) O(1) O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(m+n)BinomialHeap O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(log(n))FibonacciHeap O(1) O(log(n))* O(1)* O(1) O(log(n))* O(1)

Graphs

Node/EdgeManagement Storage AddVertex AddEdge RemoveVertex RemoveEdge QueryAdjacencylist O(|V|+|E|) O(1) O(1) O(|V|+|E|) O(|E|) O(|V|)Incidencelist O(|V|+|E|) O(1) O(1) O(|E|) O(|E|) O(|E|)Adjacencymatrix O(|V|^2) O(|V|^2) O(1) O(|V|^2) O(1) O(1)Incidencematrix O(|V||E|) O(|V||E|) O(|V||E|) O(|V||E|) O(|V||E|) O(|E|)

BigOComplexityChart



Contributors

Editthesetables!

1. EricRowell2. QuentinPleple3. NickDizazzo4. MichaelAbed5. AdamForsyth6. JayEngineer7. JoshDavis8. makosblade9. AlejandroRamirez

10. JoelFriedly11. RobertBurke12. DavidDorfman13. EricLefevreArdant14. ThomasDybdahlAhle

AROUNDTHEWEB WHAT'STHIS?



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240Comments BigOCheatSheet Login

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MichaelMitchell 2yearsagoThisisgreat.Maybeyoucouldincludesomeresources(linkstokhanacademy,moocetc)thatwouldexplaineachoftheseconceptsforpeopletryingtolearnthem.

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AmandaHarlin 2yearsago>MichaelMitchellYes!Please&thankyou

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CamCecil 2yearsago>MichaelMitchellThisexplanationin'plainEnglish'helps:http://stackoverflow.com/quest...

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ArjanNieuwenhuizen 2yearsago>MichaelMitchellHerearethelinksthatIknowof.

#1)http://aduni.org/courses/algor...#2)http://ocw.mit.edu/courses/ele...#3)https://www.udacity.com/course...

probablyasgoodormaybebetter#2,butIhavenothadachancetolookatit.http://ocw.mit.edu/courses/ele...

Sincerely,Arjan

p.s.

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p.s.https://www.coursera.org/cours...Thiscoursehasjustbegunoncoursera(dated1July2013),andlooksverygood.

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fireheron ayearago>ArjanNieuwenhuizenThankyouArjan.Espaeciallythecoursera.orgone)

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BlakeJennings 2yearsagoi'mliterallycrying

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JonRenner 2yearsagoThisisgod'swork.

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AdamHeinermann 2yearsagoIsthereaprinterfriendlyversion?

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ThomasFeichtinger 9monthsago>AdamHeinermannActuallycopyingthecontentstoagoogledocworkedprettywell!

Ihavemadeitpublic,havealook:https://docs.google.com/spread...

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ericdrowell ayearagoMod >AdamHeinermannnotyet,butthat'sagreatidea!

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MattLabrum adayago>AdamHeinermannI,too,wantedaprinterfriendlyversionforstudyingbeforeaninterview,andIwasn'tsatisfiedwiththesolutionsIfoundprovidedinthevariouscommentshere.So,IwentaheadandLaTeX'edthispagetogetanicePDF.

IhaveuploadedthePDFsIcreatedtomyGoogleDriveandmadethempublic:https://drive.google.com/folde....InthatfolderaretwoPDFSoneisforlettersizedpaperandtheotherisforA4sizedpaper.AssumingIdidn'tintroduceanytypos,thecontentofthosePDFsshouldmatchthecontentofthispage(asitappearsatthismoment17February2015),withtheonlynoteworthydifferencebeingthatImovedtheGraphssectiontobeaftertheSortingsectiontohelpeliminatesomeextrawhitespace.

JoeGibson 20hoursago>MattLabrumMatt,

GreatjobontheLaTeXdocument.I'mpreparingforaGoogleinterviewandthiswillbealotofhelp!

Anychanceyoucanputthe.texfileonyourdriveaswellinthesamefolder?

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Reply Anychanceyoucanputthe.texfileonyourdriveaswellinthesamefolder?

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MattLabrum 19hoursago>JoeGibsonDonethetwo.texfilesandthe.epsofthegrapharenowinthatfolder.

EditI'vealsoputtheRscriptandtmp.texfileIusedtocreatethegraphinthatfolder.Aftercreatingthe.epsfilewithR,IdidsomeprocessingonittogetthefinalBigO.epsfileIincludeinthe.texfiles.

Forcompleteness,togetfromtheRgeneratedBigO.epsfiletothefinalBigO.epsfile,Ididthefollowing:1.OpenBigO.epswithatexteditortoensurethetextannotationshavenotbeenbrokenapart.Ipersonallyhadtoput"Operations"and"BigOComplexity"(yaxislabelandgraphtitle)backtogether.2.Processtmp.textogeta.dvifilethatcontainsaPSFrag'edversionofthegraph.3.dvipsjEtmp.dvioBigO.tmp.eps4.epstoolcopybboxBigO.tmp.epsBigO.eps5.rmBigO.tmp.eps

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JoeGibson 19hoursago>MattLabrumThanks,yourock.

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DarrenLeRedgatr 2yearsagoIcameherefromanidletwitterclick.Ihavenoideawhatit'stalkingaboutoranyofthecomments.ButIlovethefacttherearepeopleouttherethisclever.Makesmethinkthatonedayhumanitywillcomegood.Cheers.

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GokceToykuyu 2yearsagoCouldweaddsometreealgorithmsandcomplexities?Thanks.IreallyliketheRedBlacktrees)

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ericdrowell 2yearsagoMod >GokceToykuyuExcellentidea.I'lladdasectionthatcomparesinsertion,deletion,andsearchcomplexitiesforspecificdatastructures

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ValentinStanciu 2yearsago1.Deletion/insertioninasinglelinkedlistisimplementationdependent.Forthequestionof"Here'sapointertoanelement,howmuchdoesittaketodeleteit?",singlelinkedliststakeO(N)sinceyouhavetosearchfortheelementthatpointstotheelementbeingdeleted.Doublelinkedlistssolvethisproblem.2.Hashescomeinamillionvarieties.HoweverwithagooddistributionfunctiontheyareO(logN)worstcase.Usingadoublehashingalgorithm,youendupwithaworstcaseofO(loglogN).3.Fortrees,thetableshouldprobablyalsocontainheapsandthecomplexitiesfortheoperation"GetMinimum".

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AlexisMas 6monthsago>ValentinStanciu

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AlexisMas 6monthsago>ValentinStanciuIfyoualist:ABCD,WhenyouwanttodeleteB,youcandeleteanodewithoutiteratingoverthelist.

1.B.data=C.data2.B.next=C.next3.deleteC

Ifyoucan'tcopydatabetweennodesbecauseitstooexpensivethenyes,it'sO(N)4

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Miguel 2monthsago>AlexisMasYoustillhavetofindthepositioninthelist,whichcanonlybedonelinearly.

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Guest 12daysago>MiguelYoustillhavetofindthepositioninthelist,whichcanonlybedonelinearly.

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AlexisMas 9daysago>MiguelYesofcourse,Ifyouneedtosearchthenodeit'sO(n),otherwiseyoucandeleteitasIstatedbefore.

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Guest 9daysago>AlexisMasNoneedtofindthepositionifyoucandeleteitasAlexismentioned

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OmegaNemesis28 9daysago>AlexisMasTogettoByouHAVEtoiterateoverthelistthough.Youcan'tjustmanipulateBwithoutapointer.Sounlessyoudobookkeepingandhavepointerstospecificnodesyouintendtodelete/manipulate,LinkListsareO(n)insertanddelete.

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AlexisMas 8daysago>OmegaNemesis28Strictlyspeakingno,youdon't.let'ssayyouhavethisfunction.

publicvoiddelete(Nodenode)

Thatfunctiondoesn'tcarehowdidyougotthatnode.

Didyougotmypoint?

Whenyouhaveapointertoanode,andthatnodeneedstobedeletedyoudon'tneedtoiteraveoverthelist.

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OmegaNemesis28 8daysago>AlexisMasButthatisMYpoint:p

YouhavetohavethenodeFIRST.Youhavetoiteratethroughthelistbeforeyoucandothat,unlessyoudobookkeepingandhappentohavesaidnode.RereadwhatIsaid."havepointerstospecificnodes"Mostofthetime,youdonotwithLinkedLists.Ifyou

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"havepointerstospecificnodes"Mostofthetime,youdonotwithLinkedLists.IfyouhaveaLinkedListandwanttodeleteindex5,youhavetoiterateto5andsuch.YourexamplewasABCD,ourpointsarethatyoutypicallydon'thavethepointertoBjustoffhand.YouhavetoobtainitfirstwhichwillbeO(n)

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PinguApp 3daysago>AlexisMasWhatifBisthelastelementinthelist?HowwouldB'spredecesorknowthatitsnextfieldshouldpointtoNULLandnottoafuturelyinvalidmemoryaddress?

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AlexisMas 3daysago>PinguAppInthatcaseyoucan'tdeletedthatway,you'reforcedtohaveapointertothepreviousitem.

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qwertykeyboard 2yearsagoItwouldbeveryhelpfultohaveexporttoPDF.Thx

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Gene ayearago>qwertykeyboardYoucouldconvertthedocumentyourselfusingPandoc:http://johnmacfarlane.net/pand...Itmighttakeyoualongtimetogetitworking,butPandocisanamazingonestopshopforfileconversion,andcrossplatformcompatible.IfIunderstandbigohnotationcorrectlyImightsay"IestimateyourlearningrateforlearningPandocwillbeO(1).".

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Ashutosh ayearago>GeneprovedO(n),n=numberofformatconversionstolearn:)

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JuanCarlosAlvarez ayearago>Genebigoh.hahafunny.

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Guest 2yearsagoFinalsarealreadyover...Thisshouldhavebeensharedaweekago!Wouldhavesavedmelike45minutesofusingWikipedia.

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sigmaalgebra 2yearsagoYouomittedaninplace,guaranteedO(nlog(n))arraysort,e.g.,heapsort.Youomittedradixsortthatcanbefaster

thananyofthealgorithmsyoumentioned.

MightmentionSATandrelatedproblemsinNPcomplete

wherethebestknownalgorithmforaproblemofsizenhasO(2^n).

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aproblemofsizenhasO(2^n).

Mightincludeanactual,precisedefinitionofO().9

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JonRenner 2yearsagoAnywayIcangetaPDFversionwithouttakingscreenshotsmyself?

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AttilaOlh amonthago>JonRennerPrintDestination:ChangeSelect"SaveasPDF"(inChrome).

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AntoineGrondin 2yearsagoIthinkDFSandBFS,underSearch,wouldbemoreappropriatelistedasGraphinsteadofTree.

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ericdrowell 2yearsagoMod >AntoineGrondinFixed!Thanks

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QuentinPlepl 2yearsago>AntoineGrondinAgreed

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tempire 2yearsagoThischartseemsmisleading.BigOisworstcase,notaveragecase~isaveragecase.O(...)shouldn'tbeusedintheaveragecasecolumns.

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guest 2yearsago>tempireIthinkbigOisjustanupperbound.Itcouldbeusedforall(best,worstandaverage)cases.AmIwrong?

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Luis 2yearsago>guestYouareright.

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Oleksandr ayearago>Luis@[email protected],average,andupperbound..BigO(Omicron)istheWorstCaseScenario.Itistheupperboundforforthealgorithm.Forinstanceinalinearsearchalgorithm,worstcaseiswhenthelistiscompletedoutoforder,i.e.thelistsortedbutbackwards.Omegaisthelowerbound.Thisisalmostpointlesstohave,forinstanceyouwouldratherhaveBigOthenOmegabecauseitisexactlythesameassay"itwilltakemorethanfivedollarstogettoN.Y.vs.Itswillalwaystake,atmost,135dollarstogettoNewYork."ThefirstbitofinformationfromOmegaisessentiallyuseless,thethirdhowevergivesyoutheconstraint.Thetaistheupperandlowerboundtogether.Thisisthemostbeneficiallypieceofinformationtohaveaboutanalgorithmbutunfortunatelyitisusuallyveryhardto

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find,butwehavedonethis.Youcanusuallyfindthataverageforanalgorithmsefficiencybytestingitinaveragecaseandworstcasestogether.Simplythisisacomputationalexercisetoextracttheempiricaldata.ThereisanotherproblemIdonotlikeisthecolorschemeissometimeswrong..O(n)isbettertheO(log(n))?Inwhatway?1024vs10incrementsthatasortalgorithmhastoperformforinstance?Allinallthisisgoodinformationbutinitscurrentstate,tothenovice,honestlyitneedstobetakenwithagrainofsaltandfactcheckwithagoodalgorithmbook.However,thisisinMHOsoifI'moffbaseorincorrectthenfeelfreetoflamemelikethefantasticfouratagayparade:)

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Guest ayearago>Oleksandr@OleksandrYouareconfused.Yourexampleaboutthedollarsstatesspecificamounts(e.g."atmost135dollars"),butbigOandrelatedconceptsareusedtoboundtheorder(linear,exponential,etc.)ofafunctionthatdescribeshowanalgorithmgrows(inspace,time,etc.)withproblemsize.Tobemoreappropriate,yourexampleshouldbemodifiedtosaysomethinglike"ittakesatmost2$permile"(linear).Withthisinmind,youcanthusunderstandhowbigOcanbeusedbothfor,say,thebestandtheworstcase.Takeyourlinearsearch.Asthesizeoftheproblemgrows(thearraytobesearchedgrowsinsize),thebestcasestillhasanuppertimeboundofO(1)(ittakesconstanttimetofindanelementinindex0,oranotherfixedposition),whiletheworstcase(theobjectisinthelastindexwherewelook)hasanuppertimeboundofO(n)(ittakesanumberofstepsoforderequaltotheproblemsize,n,untilwefindtheobjectinthelastindexwherewelook.).

(fixed:wrongautocompleteofwhoIrepliedto)3

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PhilipMachanick ayearago>OleksandrOmegaisuselessunlessitisatightbound,i.e.,itrepresentsrealminimalcasesthatareinteresting(whenyouhaveoptionslike=inthedefinitionofabound,youshouldatleastgetclosetothe=case,otherwiseyoumightaswellusethestrictlycases,andeventhereyoushouldtrytofindboundsthatarereasonablyclosetothe=case).Forexample,strictlyspeaking,quicksortisOmega(1),butOmega(nlogn)ismoreinformativebecausetellsyouitsrealbestcase.

Inanycase,youdonotnormallyuseOmega,Thetaetc.fordifferentiatingaverage,bestandworstcase.Theseareboundsonanyofthesecases.Forquicksort,theworstcaseanalysisisn^2andthisisboththeupperandlowerboundontheworstcase.YouuseOmega,Theta,etc.whentheanalysisforaparticularcaseisnotclearandyouhavetosayitisnobetterthanornoworsethanaparticularanalysis.

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Luis ayearago>Oleksandr@OleksandrYouareconfused.Yourexampleaboutthedollarsstatesspecificamounts(e.g."atmost135dollars"),butbigOandrelatedconceptsareusedtoboundtheorder(linear,exponential,etc.)ofafunctionthatdescribeshowanalgorithmgrows(inspace,time,etc.)withproblemsize.Tobemoreappropriate,yourexampleshouldbemodifiedtosaysomethinglike"ittakesatmost2$permile"(linear).Withthisinmind,youcanthusunderstandhowbigOcanbeusedbothfor,say,thebestandtheworstcase.Takeyourlinearsearch.Asthesizeoftheproblemgrows(thearraytobesearched

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Takeyourlinearsearch.Asthesizeoftheproblemgrows(thearraytobesearchedgrowsinsize),thebestcasestillhasanuppertimeboundofO(1)(ittakesconstanttimetofindanelementinindex0,oranotherfixedposition),whiletheworstcase(theobjectisinthelastindexwherewelook)hasanuppertimeboundofO(n)(ittakesanumberofstepsoforderequaltotheproblemsize,n,untilwefindtheobjectinthelastindexwherewelook.).

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Oleksandr1 ayearago>LuisYoumakeaverypoorassumptionthatbecauseaspecificvalueisgiven,thanitmustbealinearfunction.ItisinfactanypolynomialfunctionofmychoicegivenitsparametersandanyamountofLagrangeconstants

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big-o algorithm complexity cheat sheet

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