Ukuhlalutya i-Algorithm Complexity Umahluko weMTSC7196

Ukuqonda i-Algorithm Complexity

Ixesha vs. Ukuntsonkotha kwesithuba

Ubunzima be-algorithm ngokuyintloko kujongana nezixhobo ezimbini: ixesha (ixesha lokwenziwa) kunye indawo (ukusetyenziswa kwememori). Ngelixa ixesha elintsonkothileyo lilinganisa ukuba ixesha lokubaleka likhula njani ngobungakanani begalelo ( n ), ubunzima bendawo buvavanya ukusetyenziswa kwememori. Umzekelo:
- I-algorithm ene O(n) ixesha elintsonkothileyo lilinganisa ngokobungakanani begalelo.
- I-algorithm ene O(1) indawo entsonkothileyo isebenzisa inkumbulo engaguqukiyo nokuba ingakanani na igalelo.

Zombini iimetriki zibalulekile. I-algorithm ekhawulezayo inokukhupha imemori kwiiseti zedatha ezinkulu, ngelixa i-algorithm esebenza kakuhle imemori inokucotha kakhulu kwizicelo zexesha lokwenyani.

Ukubaluleka kuYilo lwe-algorithm

Ukusebenza kakuhle kumisela ukuba nokwenzeka. Cinga ngokuhlela uluhlu lwezinto ezili-10 ngokuchasene ne-10 lezigidi:
- A uhlobo lweqamza ( O(n) ) inokwanela iseti yedatha encinci kodwa ayinakwenzeka kwezinkulu.
- A dibanisa uhlobo ( O(n log n) ) iphatha iiseti zedatha ezinkulu ngobubele kodwa ifuna inkumbulo eyongezelelweyo.

Uhlalutyo oluntsonkothileyo lubonelela ngolwimi lwehlabathi lonke ukuthelekisa i-algorithms, kutsalwa kude iinkcukacha ezithe ngqo zehardware. Ixhobisa abaphuhlisi ukuba baqikelele i-scalability kwaye baphephe imiqobo kwiinkqubo ezibalulekileyo.

Iingcaphuno ze-Asymptotic: Ulwimi oluntsonkothileyo

Ubhalo lwe-Asymptotic luchaza indlela yokuziphatha ethintelayo yemisebenzi, enikezela ngokufutshane ngobunzima. Iimpawu ezintathu eziphambili:

UBig O (O): Upper Bound (Eyona meko imbi)

I-Big O notation ichaza elona xesha liphezulu okanye isithuba esithathwa yi-algorithm. Njengokuba:
- O(1) : Ixesha elithe rhoqo (umzekelo, ukufikelela kuluhlu ngokwesalathiso).
- O(n) : Ixesha lomgca (umzekelo, ukuphinda-phinda kuluhlu).
- O(n) : Ixesha lequadratic (umzekelo, iirhintyela ezifakwe kwindlwane ngohlobo lwamaqamza).

IBig O yeyona metric isetyenziswa kakhulu, njengoko iqinisekisa iisilingi zokusebenza.

Omega : Umda osezantsi (Owona mzekelo)

I-Omega ichaza ubuncinane bexesha elifunekayo. Umzekelo:
- Ukukhangela ngomgca (1) ukuba ekujoliswe kuko sisiqalelo sokuqala.

Ngelixa kunethemba, uhlalutyo lwemeko engcono kakhulu alunalwazi kangako kwisicwangciso semeko embi kakhulu.

Theta : Ibotshelelwe ngokuqinileyo (I-Avareji-Case)

I-Theta idibanisa i-Big O kunye ne-Omega, emele indlela yokuziphatha engabonakaliyo. Ukuba i-algorithms eyona nto ingcono kunye neemeko ezimbi ziyafana:
- (n log n) Isebenza ekudityanisweni kweentlobo zomndilili kunye neemeko ezimbi kakhulu.

Olu bhalo luyimiba engaguqukiyo kunye nemigaqo ephantsi, igxile kumazinga okukhula. Njengokuba, 2n + 3n + 4 yenza lula uku O(n) kuba igama lequadratic lilawula ubukhulu n .

Iiklasi eziQhelekileyo ezintsonkothileyo

Ukuqonda iiklasi ezintsonkothileyo kunceda ukwahlula ii-algorithms ngokukala. Nalu uluhlu lwemigangatho ukusuka kweyona incinci ukuya kweyona isebenzayo:

O(1): Ixesha elithe rhoqo

Ixesha lokwenziwa okanye inkumbulo ihlala ingatshintshwanga n iyakhula.
- Umzekelo : Ukufikelela kwixabiso letafile ye-hash ngesitshixo.

O(log n): Ixesha leLogarithmic

Ixesha lokubaleka likhula ngokwe-logarithmically nge n .
- Umzekelo : Uphendlo lwe-binary lisiqingatha isithuba songeniso ngokuphinda-phinda.

O(n): Ixesha lomgca

Ixesha lokubaleka lilinganisa ngokulinganayo nge n .
- Umzekelo : Uphendlo lomgca kuluhlu olungahlelwanga.

O(n log n): Ixesha lelinearithmic

Ixhaphakile ekwahluleni-kwaye-koyisa i-algorithms.
- Umzekelo : Dibanisa uhlobo kunye nohlobo lwemfumba.

O(n): Ixesha lequadratic

Uphindaphindo olwenziwayo lukhokelela ekukhuleni okugqabhukileyo.
- Umzekelo : Uhlobo lwebhamuza kunye nohlobo lokhetho.

O(2): Ixesha elililo

Ixesha lokubaleka liphinda kabini ngegalelo ngalinye elongezelelweyo.
- Umzekelo : Ukubala okuphindiweyo kweFibonacci ngaphandle kokukhumbula.

O(n!): Ixesha leFactorial

I-algorithms esekwe kwimvume.
- Umzekelo : Ukusombulula ingxaki yomthengisi ohambahambayo nge-brute-force.

Umahluko phakathi O(n log n) kwaye O(n) iba stark for n = 10 : yangaphambili inokuphumeza kwi-milliseconds, ngelixa ukugqibela kunokuthatha iintsuku.

Uhlahlelo lweCase: Eyona nto iPhambili, i-Avareji, kunye neyona meko imbi kakhulu

Ii-algorithms zisebenza ngokwahlukileyo ngokusekelwe kuqwalaselo lwegalelo. Ukuhlalutya zonke iimeko kuqinisekisa ukomelela:

Eyona meko igqwesileyo: Igalelo Elilelona

• Umzekelo : QuickSorts inyathelo lokwahlulahlula uluhlu ngokulinganayo, lunika O(n log n) .

Eyona nto imbi kakhulu: Igalelo lePathological

• Umzekelo : QuickSort yehlisa ukuya O(n) ukuba i-pivot yeyona nto incinci kuluhlu oluhleliweyo.

I-Avareji-Case: Ungeniso olungephi

• Umzekelo : I-avareji ye-QuickSort O(n log n) kwidatha engalungiswanga.

Iimpembelelo Eziluncedo

Isilungisi sombuzo wesiseko sinokukhetha phakathi kodibaniso lwe-hash ( O(n + m) ) kunye nendibano ye-loop enendlu ( O(nm) ) ngokusekelwe ekusasazeni idatha. Uhlalutyo olubi kakhulu lubalulekile kwiinkqubo ezibaluleke kakhulu zokhuseleko (umzekelo, isofthiwe ye-aviation), apho ukungaqiniseki kungamkelekanga.

Ukuthelekisa ii-algorithms zeNgxaki efanayo

Ingxaki efanayo ingasonjululwa kusetyenziswa iindlela ezahlukeneyo zokuziqhelanisa. Umzekelo, ingxaki yokukhangela ixabiso ekujoliswe kulo kuluhlu lwamaxabiso inokusonjululwa kusetyenziswa iindlela ezahlukeneyo zokuziphatha, ezifana nophendlo lomgca, uphendlo lokubini, okanye uphendlo lwetafile yehash.

Itheyibhile engezantsi ithelekisa ixesha kunye nobunzima bendawo yale algorithms yokukhangela ixabiso ekujoliswe kulo kuluhlu lwe n ixabiso.

Ukukhethwa kwe-algorithm kuxhomekeke kubungakanani bengxaki, iimpawu zegalelo, kunye nezixhobo ezikhoyo. Umzekelo, ukuba uluhlu luncinci kwaye aluhlelwanga, ukukhangela ngemigca inokuba lolona khetho lulungileyo. Ukuba uluhlu likhulu kwaye luhleliwe, uphendlo lokubini lunokuba lolona khetho lungcono. Ukuba uluhlu lukhulu kwaye aluhlelwanga, ukukhangela kwetafile ye-hash kunokuba lolona khetho lulungileyo.

Imixholo ekwiNgxaki yokuHlalutya

Uhlalutyo lwe-Amortized

Uhlalutyo lwe-amortized iavareji yexesha kulandelelwano lwemisebenzi.
- Umzekelo : Uluhlu olunamandla oluphindwe kabini xa lugcwele. Ngelixa eyedwa Dudula utyando lunokuthatha O(n) ixesha, iindleko amortized uhlala O(1) .

Uhlalutyo olunokwenzeka

Algorithms ezifana Monte Carlo kwaye Las Vegas sebenzisa i-random ukuze usebenze kakuhle.
- Umzekelo : Uvavanyo lweprimality lweMiller-Rabin luneziqinisekiso ezinokubakho kodwa lukhawuleza kuneendlela zokuqinisekisa.

I-NP-UkuGcwalisa kunye nokuNcitshiswa

Ezinye iingxaki (umzekelo, ukwaneliseka kweBoolean) zi I-NP-igqityiwe , okuthetha ukuba akukho sisombululo saziwayo sexesha lepolynomial sikhoyo. Ukubonisa ukugqibelela kwe-NP ngokunciphisa kunceda ekuhleleni ubunzima bokubala.

Iimpembelelo eziSebenzayo zoMahluko oManziyo

IDatha enkulu kunye nokuFunda koomatshini

An O(n) ialgorithm yokudibanisa inokuba ngumqobo kwiiseti zedatha ezinkulu, ikhuthaza utshintsho kwiindlela eziqikelelweyo njengemithi ye-kd ( O(n log n) ).

Ubhalo ngokufihlakeleyo

Iinkqubo zesitshixo zikawonke-wonke zithembele kubulukhuni be O(2) iingxaki (umzekelo, i-integer factorization) ukuxhathisa uhlaselo.

Uphuhliso lomdlalo

Iinjini zonikezelo lwexesha lokwenyani zibeke phambili O(1) Ii-algorithms zokulinganisa iifiziksi ukugcina i-60+ FPS.

Ukukhetha i-algorithm eLungileyo

Urhwebo lubalulekile:
- Ixesha vs. Isithuba : Sebenzisa iimephu zehash ( O(1) lookups) ngexabiso lememori.
- Ubulula vs. Ukuba nempumelelo : uhlobo lofakelo ( O(n) ) isenokukhethwa kwiseti yedatha encinci, ephantse yahlelwa.

Izixhobo kunye noBuchule bokuHlalutya ukuntsonkotha

Ubudlelwane obuphindaphindiweyo

Kwii-algorithms eziphindaphindayo, imodeli yobudlelwane obuphindaphindiweyo ngexesha lokusebenza. Umzekelo, ukudibanisa iintlobo ukuphindaphinda:
[ T (n) = 2T(n/2) + O(n) ] ugqiba ekubeni O(n log n) nge Ithiyori yoMphathi .

Ukuthelekisa

Uvavanyo lobugcisa luncedisa uhlalutyo lwethiyori. Izixhobo zokuchaza iinkcukacha (umzekelo, i-Valgrind, i-perf) zityhila iimeko zehlabathi zokwenyani.

Uhlalutyo lwe-Asymptotic kwiKhowudi

inhlwathi

O(n) ukuntsokotha kwexesha

def linear_sum(arr):
iyonke = 0
nge num in arr:
iyonke += inani
imbuyekezo iyonke

O(n) ukuntsokotha kwexesha

def quadratic_sum(arr):
iyonke = 0
kuba ndihleli:
kuba j in arr:
iyonke += mna * j
imbuyekezo iyonke

Imigibe eqhelekileyo kunye neengcamango ezingezizo

Ukungahoyi i-Constants kunye ne-Lower-Order Terms

Ngelixa O(n) abstracts kude constants, a 100n I-algorithm inokuba kade kuno-a 0.01n algorithm for practical n .

Ukujongwa ngendlela engeyiyo ubungakanani begalelo

An O(n log n) I-algorithm inokuthi isebenze ngaphantsi O(n) ngenxa n = 10 ngenxa yobunzima.

Ukujongana nobunzima beSithuba

Umsebenzi owenziwe ngentloko weFibonacci ( O(n) indawo) inokuntlitheka kumagalelo amakhulu, ngokungafaniyo noguqulelo oluphindaphindayo ( O(1) indawo).

Ukubhidanisa eyona meko imbi kunye ne-Avareji-Case

I-BST yokuzilinganisa ( O(log n) Ukukhangela) kukhuselekile kune-BST eqhelekileyo ( O(n) eyona nto imbi) kwidatha engathenjwa.

Ukuqukumbela

Uhlalutyo oluntsonkothileyo lwe-algorithm yikhampasi ekhokela abaphuhlisi kumhlaba omkhulu wobuchule bokubala. Kubafundi be-MTSC7196, ukugqwesa olu qeqesho kudibanisa ulwazi lwethiyori kunye nobuchule obusebenzayo. Ngokucazulula iimfuno zexesha kunye nendawo, ukuthelekisa imida ye-asymptotic, kunye nokuhamba ngezorhwebo lehlabathi langempela, abaphuhlisi banokwenza iinkqubo zokwenza isikali esilungileyo kwaye senze ngokuthembekileyo.

Ngexesha elichazwe yidatha-eqhutywa ngokutsha, ukukwazi ukwahlula phakathi kwe O(n log n) kunye ne O(n) Isisombululo ayisosinyanzelo nje semfundo ephakamileyo. Njengoko uqhubela phambili kwizifundo zakho, khumbula: uhlalutyo lokuntsokotha alukho nje ngamanani kunye neesimboli. Imalunga nokuqonda ukubetha kwentliziyo yokubala ngokwayo.