Yin Nazari Bambance-bambancen Hadarin Algorithm na MTSC7196

Fahimtar Haɗin Algorithm

Lokaci vs. Complexity na sararin samaniya

Matsalolin algorithm da farko yana magance albarkatu biyu: lokaci (lokacin aiwatarwa) da sarari (amfani da memory). Yayin da rikitarwar lokaci ke auna yadda lokacin aiki ke girma tare da girman shigarwa ( n ), Rukunin sararin samaniya yana kimanta amfani da ƙwaƙwalwar ajiya. Misali:
- algorithm tare da O(n) rikitaccen lokaci yana daidaita ma'auni a layi tare da girman shigarwar.
- algorithm tare da O(1) rikitaccen sararin samaniya yana amfani da ƙwaƙwalwar ajiya akai-akai ba tare da la'akari da girman shigarwar ba.

Duk ma'auni biyun suna da mahimmanci. Algorithm mai sauri na iya ƙyale ƙwaƙwalwar ajiya akan manyan ma'ajin bayanai, yayin da ingantaccen tsarin ƙwaƙwalwar ajiya zai iya zama jinkirin yin aikace-aikacen ainihin lokaci.

Muhimmanci a Tsarin Algorithm

Inganci yana nuna yiwuwa. Yi la'akari da rarraba jerin abubuwa 10 da miliyan 10:
- A nau'in kumfa ( O(n) ) zai iya isa ga ƙananan bayanan bayanai amma ya zama mara amfani ga manya.
- A hade iri ( O(n log n) ) yana sarrafa manyan bayanan bayanai da alheri amma yana buƙatar ƙarin ƙwaƙwalwar ajiya.

Binciken rikitarwa yana ba da harshe na duniya don kwatanta algorithms, yana kawar da takamaiman bayanai na hardware. Yana ba wa masu haɓaka damar yin hasashen haɓakawa da kuma guje wa ɓarna a cikin m tsarin.

Bayanin Asymptotic: Harshen Rudani

Bayanan asymptotic suna bayyana ƙayyadaddun halayen ayyuka, suna ba da gajeriyar hannu don rikitarwa. Bayanan farko guda uku sune:

Babban O (O): Babban Bound (Mafi Muni)

Babban bayanin kula yana bayyana iyakar lokaci ko sarari da algorithm zai ɗauka. Misali:
- O(1) : Matsakaicin lokaci (misali, samun dama ga sashin tsararru ta fihirisa).
- O(n) : Lokacin layin layi (misali, maimaita ta hanyar jeri).
- O(n) : Lokacin ma'auni (misali, madaukai na gida a cikin nau'in kumfa).

Big O shine ma'aunin da aka fi amfani dashi, saboda yana ba da garantin aikin silin.

Omega: Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarƙashin Ƙarfi (Mafi Mafi Girma).

Omega ya bayyana mafi ƙarancin lokacin da ake buƙata. Misali:
- Binciken layi yana da (1) idan makasudin shine kashi na farko.

Yayin da ake da kyakkyawan fata, mafi kyawun bincike ba shi da fa'ida don tsara mafi munin yanayi.

Theta: Tight Bound (Matsakaici-Case)

Theta ya haɗa Big O da Omega, yana wakiltar ainihin halin asymptotic. Idan algorithms mafi kyau da mafi munin lokuta iri ɗaya ne:
- (n log n) ya shafi haɗa matsakaicin matsakaici da mafi munin yanayi.

Waɗannan bayanin kula suna kawar da madaidaicin ƙima da ƙarancin tsari, suna mai da hankali kan ƙimar girma. Misali, 2n+3n+ 4 sauƙaƙa zuwa O(n) saboda kalmar quadratic ta mamaye babba n .

Darussan Complexity gama gari

Fahimtar azuzuwan masu rikitarwa yana taimakawa rarrabuwar algorithm ta hanyar scalability. Anan ga matsayi daga mafi ƙarancin inganci zuwa mafi ƙarancin inganci:

O(1): Tsawon Lokaci

Lokacin aiwatarwa ko ƙwaƙwalwar ajiya ya kasance baya canzawa kamar n girma.
- Misali : Samun damar ƙimar tebur ɗin hash ta maɓalli.

O(log n): Logarithmic Time

Lokacin gudu yana girma logarithmically tare da n .
- Misali : Binciken binary yana rage rabin wurin shigarwa kowane juzu'i.

O(n): Lokacin Litattafai

Lokacin gudu yana daidaita daidai da n .
- Misali : Binciken layi ta hanyar lissafin da ba a ware ba.

O(n log n): Lokacin Linearithmic

Na kowa a cikin algorithms rabo-da-ci.
- Misali : Haɗa nau'i da nau'in tsiri.

O(n): Lokaci Quadrate

Ƙirar gida tana haifar da girma mai fashewa.
- Misali : nau'in kumfa da nau'in zaɓi.

O(2): Tsawon Lokaci

Lokacin gudu yana ninka tare da kowane ƙarin shigarwar.
- Misali : Recursive Fibonacci lissafin ba tare da hadda.

O(n!): Lokacin Factoryal

Algorithms na tushen permutation.
- Misali : Magance matsalar mai siyar da balaguro ta hanyar baƙar fata.

Bambanci tsakanin O(n log n) kuma O(n) ya zama mai kauri don n = 10 : na farko zai iya aiwatarwa a cikin millise seconds, yayin da na ƙarshe zai iya ɗaukar kwanaki.

Binciken Harka: Mafi Kyau, Matsakaici, da Mafi Mummunan Hali

Algorithms suna yin daban-daban dangane da saitunan shigarwa. Yin nazarin duk lamuran yana tabbatar da ƙarfi:

Mafi kyawun Harka: Mafi kyawun shigarwa

• Misali Mataki na bangare na QuickSorts yana raba tsararru a ko'ina, yana ba da kyauta O(n log n) .

Mafi Muni: Input na Pathological

• Misali : QuickSort ya ƙasƙanta zuwa O(n) idan pivot shine mafi ƙanƙanci a cikin tsararrun tsararru.

Matsakaicin-Case: Bazuwar Input

• Misali Matsakaicin QuickSort O(n log n) don bayanan da ba a daidaita su ba.

Tasirin Aiki

Mai inganta binciken bayanai na iya zaɓar tsakanin haɗin hash ( O(n +m) ) da madauki madauki ( O(nm) ) bisa ga rarraba bayanai. Binciken mafi muni yana da mahimmanci ga tsarin aminci-mafi mahimmanci (misali, software na jirgin sama), inda ba za a yarda da tsinkaya ba.

Kwatanta Algorithms don Matsala iri ɗaya

Ana iya magance wannan matsala ta amfani da algorithms daban-daban. Misali, ana iya magance matsalar neman kimar manufa a cikin jerin ƙididdiga ta amfani da algorithms daban-daban, kamar binciken layi, binciken binary, ko binciken tebur na zanta.

Teburin da ke ƙasa yana kwatanta hadaddun lokaci da sararin samaniya na waɗannan algorithms don neman ƙimar manufa a cikin jerin n dabi'u.

Zaɓin algorithm ya dogara da girman matsalar, halayen shigarwa, da albarkatun da ake da su. Misali, idan jerin ƙananan ne kuma ba a daidaita su ba, bincike na layi yana iya zama mafi kyawun zaɓi. Idan jerin suna da girma kuma an jera su, binciken binary na iya zama mafi kyawun zaɓi. Idan jeri ne babba kuma ba a daidaita shi ba, binciken tebur na zanta na iya zama mafi kyawun zaɓi.

Batutuwa Masu Cigaba a cikin Tattalin Arziki

Amortized Analysis

Amortized bincike yana da matsakaicin lokaci akan jerin ayyuka.
- Misali : Tsarukan daɗaɗɗen iya aiki sau biyu idan ya cika. Yayin daya tura aiki na iya ɗauka O(n) lokaci, da amortized kudin ya rage O(1) .

Binciken Mai yiwuwa

Algorithms kamar Monte Carlo kuma Las Vegas yi amfani da bazuwar don dacewa.
- Misali : Gwajin farko na Miller-Rabin yana da garantin yuwuwar amma ya fi sauri fiye da hanyoyin tantancewa.

NP-Kammalawa da Ragewa

Wasu matsalolin (misali, gamsuwar Boolean) sune NP-cikakke , ma'ana babu sanannen bayani na lokaci-lokaci. Tabbatar da cikar NP ta hanyar ragi yana taimakawa rarrabuwa taurin lissafi.

Abubuwan Aiki Na Bambance-bambancen Rukuni

Babban Bayanai da Koyan Injin

An O(n) Algorithm na tari na iya zama ƙwanƙwasa ga manyan bayanai, yana haifar da sauye-sauye zuwa kusan hanyoyin kamar bishiyoyin kd. O(n log n) ).

Rubutun Rubutu

Tsarin maɓalli na jama'a sun dogara da taurin O(2) matsaloli (misali, ƙididdiga ƙididdiga) don tsayayya da hare-hare.

Ci gaban Wasan

Injin samar da lokaci na ainihi suna ba da fifiko O(1) Algorithms don simintin physics don kula da 60+ FPS.

Zaɓin Algorithm Dama

Ciniki-offs yana da mahimmanci:
- Lokaci vs. sarari : Yi amfani da taswirar hash ( O(1) Lookups) a farashin ƙwaƙwalwar ajiya.
- Sauƙi vs. Mafi kyawu : nau'in shigar ( O(n) ) ƙila ya fi dacewa ga ƙananan, kusan tsararrun saitin bayanai.

Kayayyaki da Dabaru don Nazari Complexity

Dangantakar Maimaituwa

Don algorithms masu maimaitawa, maimaitawar dangantakar da ke gudana lokacin aiki. Misali, haɗa nau'ikan maimaitawa:
[T (n) = 2T(n/2) + O(n)] ya yanke zuwa O(n log n) ta hanyar Jagora Theorem .

Benchmarking

Gwajin ƙwaƙƙwara yana haɓaka nazarin ka'idar. Kayayyakin bayanan bayanai (misali, Valgrind, perf) suna bayyana ƙulla-ƙulle na gaske.

Asymptotic Analysis a cikin Code

Python

O(n) rikitarwa lokaci

def linear_sum(arr):
jimla = 0
don num in arr:
jimlar += lamba
dawo duka

O(n) rikitarwa lokaci

def quadratic_sum(arr):
jimla = 0
don i in arr:
za j a arr:
jimlar += i * j
dawo duka

Matsalolin gama gari da rashin fahimta

Yin watsi da Constants da Sharuɗɗan Oda Ƙananan

Yayin O(n) abstracts nesa akai akai, a 100n Algorithm na iya zama a hankali fiye da a 0.01n algorithm don m n .

Kuskure Girman Abubuwan Shigarwa

An O(n log n) algorithm iya kasa aiki O(n) domin n = 10 saboda wuce gona da iri.

Kallon Rukunin Sararin Samaniya

Aikin Fibonacci da aka haddace ( O(n) sarari) na iya yin karo akan manyan abubuwan shigar da bayanai, sabanin sigar maimaitawa ( O(1) sarari).

Rikicin Mafi Muni da Matsakaici-Case

BST mai daidaitawa ( O(log n) bincike) ya fi aminci fiye da BST na yau da kullun ( O(n) mafi munin yanayi) don bayanan da ba a amince da su ba.

Kammalawa

Binciken hadaddun algorithm shine jagorar kamfas da ke jagorantar masu haɓaka ta hanyar faffadan fage na ingancin ƙididdigewa. Ga ɗaliban MTSC7196, ƙwarewar wannan horon yana gadar ilimin ƙa'idar da ƙwarewar aiki. Ta hanyar rarraba lokaci da buƙatun sararin samaniya, kwatanta iyakokin asymptotic, da kuma kewaya kasuwancin duniya na ainihi, masu haɓakawa za su iya ƙirƙira tsarin da ke da ma'auni cikin alheri kuma suna aiki da dogaro.

A cikin wani zamani da aka siffanta ta hanyar ƙirƙira bayanai, ikon gane tsakanin wani O(n log n) kuma an O(n) Magani ba ilimi ba ne kawai mahimmancin dabara. Yayin da kuke ci gaba ta hanyar karatunku, ku tuna: rikitaccen bincike ba kawai game da lambobi da alamomi ba. Yana da game da fahimtar bugun zuciya na lissafin kanta.