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Introduction to Algorithm Analysis and Design: Lecture 1, Exams of Algorithms and Programming

Ain Shams University Algorithms and Programming

An introduction to Algorithm Analysis and Design, discussing various concepts such as algorithm analysis, resources, time and space complexity, and different types of analysis. It also includes examples and pseudo code for calculating run time for algorithms and analyzing their complexity.

Typology: Exams

2018/2019

Uploaded on 12/31/2021

hassan-amer-1 🇪🇬

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Lecture # 1

Introduction:

ضَٕ ذششنا اذثَ اي مثق ّياذتذ ّغفَ تار طسٕكنا ُٗؼي ... ًّعا طسٕكنا

Introduction to Algorithm Analysis and Design

لا ٌا ٍيفساػ اُزا ٌا ضٔشفًنا Algorithm ُّيؼي ًّٓي زيفتن ُّيؼي تإطخ ّػًٕدي ٍػ ِساثػ

لاثي ُٗؼي function لا summation آفشؼي اَا ٗهنا آتاتاغز مًؼت ائْ ٍيًقس آيدا اَا ٌا آتًٓي مك

عًٕدًنا ةغستٔ

لا فيشؼتن ّثغُنات ٌاك ِد Algorithm

مًػا اَا ٌا ُٗؼي ّيا اَا ةطAnalysis & Design for an Algorithm ؟ ِذك مًؼْ ّينٔ

ٍي ذخاي شئ غيشع ٌٕكي حياَشثنا ٌا آُي ساثتػلاا ٗف اْذخا وصلا شيتك ميإػ ّيف حياَشت ٖا مًؼت اَأ لأا

لا Resources ٌاك ءإع شيتك ٗتػاتتmemory or disk حياَشت ضيًتت شيتك ّيَات ميإػٔ لٔد ٍيهياؼنا

َٗات حياَشت ٍػ

لا ٗتفٕند ةط analysis ّيا ٍػ ِساثػ ِد

لاanalysis لات ىتٓيت performance لات لأperformance أًْ لاا ٍييغيئس ٍيقش ٍػ ِساثػ

لا Space and Time

لاspace لا ايا ّيهػ اُنٕق Time مًؼُت اُزأ ٗغيئشنا اُياًتْا ٗقثيْ ِد اف analysis ِد طسٕكنا ٗف

مًػا اَا ٌا ُٗؼيAnalysis ل Algorithm لا شثتخا اَا ٌا ٍيؼي performance ٌٕكيْ مْ ّػاتت

Efficient ؟لا لأ آيت ّهست اَا ٗهنا ّنأغًهن

ساتخا لاثي ٍكًي اَا ٌا ُٗؼًت Algorithm مًػا ٌاشػ sorting ٍكينٔ insertion sort

لاinsertion sort ٕهيدا اًن input لا ٍي شيغص array ٕن ُٗؼي array with size 20 واًت مغتشيْ

مًؼت اَا ٕن ٍكنapplication لأ شيثك size لا عاتت array ّػشغت مغتشيْ شي ف اذخ شيثك

من ّثغُنات اياDesign

لا مًؼَ ٖاصا ىهكتيت ْٕ افAlgorithm ٌٕكي ٗهنا efficient for my program

ّياػ ّفصت طسٕكهن ّثغُنات ِد..

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Lecture # 1

Introduction:

Introduction to Algorithm Analysis and Design

ُّػثاس ِ ػ ٍ يدًٕػ ّ خطٕات يؼي ُّ نتفيز ي ًّٓ يؼي Algorithm انًفشٔض ا ٌ ازُا ػاسفي ٍ ا ٌ ال

كم يًٓتٓا ا ٌ اَا اديٓا سقًي ٍ ْٔيا تؼًم زغاتاتٓا انه ٗ اَا يؼشفٓا summation الfunction يؼ ُٗ يثال ٔتسغة انًدًٕع

Algorithm د ِ كا ٌ تانُغث ّ نتؼشيف ال

ٔني ّ ْؼًم كذ ِ ؟ Analysis & Design for an Algorithm طة اَا اي ّ يؼ ُٗ ا ٌ اَا اػًم

ٍأال ٔاَا تؼًم ا ٖ تشَايح في ّ ػٕايم كتيش الصو اخذْا ف ٗ االػتثاس يُٓا ا ٌ انثشَايح يك ٌٕ عشيغ ٔيش ياخذ ي انؼايهي ٍ دٔل ٔػٕايم تاَي ّ كتيش تتًيض تشَايح memory or disk تتاػت ٗ كتيش عٕاء كاٌ Resourcesال َٗػ ٍ تشَايح تا

ّد ِ ػثاس ِ ػ ٍ اي analysis طة دنٕفت ٗ ال

ػثاس ِ ػ ٍ شقي ٍ سئيغيي ٍاال ًْٔا performanceٔال تال performanceتيٓتى تال analysis ال

Space and Time ال

ِف ٗ انكٕسط د analysis فا د ِ ْيثق ٗ اْتًايُا انشئيغ ٗ ٔازُا تُؼًم Timeقٕنُا ػهي ّ ايا ال space ال

ٌٕتتاػ ّ ْم ْيك performance يؼي ٍ ا ٌ اَا اختثش ال Algorithm لAnalysis يؼ ُٗ ا ٌ اَا اػًم نهًغأن ّ انه ٗ اَا تسه ّ تيٓا ٔال ال؟ Efficient

insertion sort ٍٔنيك sorting ػشا ٌ اػًم Algorithm تًؼ ُٗ ا ٌ اَا يًك ٍ يثال اختاس

ْيشتغم تًاو array with size 20 ٕيؼ ُٗ ن arrayصغيش ي ٍ ال input ٕنًا اديه insertion sort ال

ّكثيش خذا ف يش ْيشتغم تغشػ arrayتتاع الsizeكثيش ٔال application نك ٍ ن ٕ اَا تؼًم

Design ايا تانُغث ّ نم

efficient for my program ٌٕانه ٗ يك Algorithm فا ْٕ تيتكهى اصا ٖ َؼًم ال

..ّد ِ تانُغث ّ نهكٕسط تصف ّ ػاي

Calculating Run Time for any Algorithm:

ٖشغال تأ Programmer ٖػهشا ٌ يقذس ا pseudo codeتيثق ٗ يكتٕب تال Algorithmٖنًا تُذسط ا د ِ ٔ يسٕنٕا نكٕد تانهغ ّ ان ٗ شغال تيٓا Algorithmا َّ يفٓى ال programming language

Algorithmد ِ اَُا َؼشف اص ٖ ال Analysisانغشض ي ٍ ال Algorithm ٖال Analysis نًا َسة َؼًم ُْٗٔؼشف دنٕقت Worst Caseتتاػتٕا ف ٗ ال run timeانًختهف ّ ٔ َؼشف ال Input د ِ تيتؼايم يغ ال ياػ ُٗ ايّ

Loops, Statements,(زاخات 4 ٍتيتك ٌٕ ي sAlgorithm ٖطية دنٕقت ٗ ا Worst Case ال .) Conditions, Recursion

دي ّ كهٓا ػه ٗ تؼط ٔ تطهؼه ٗ ف ٗ االخش Costsٔ ف ٗ االخش تدًغ كم ال Cost كم زاخ ّ ي ٍ دٔل تيثق ٗ نيٓا ان ٗ تيغتٓهكٕا ال Timeٔدي ّ تتثق ٗ ػثاس ِ ػ ٍ ال Time Complexity يؼادن ّ اعًٓا يؼادن ّ ال ِد Algorithm

N ان ٗ ْٕا Inputف ٗ ػذد ال As a Function

صغيش كذ ِ َدشب في ّ انكالو د ِ َٔشٕف انًؼادن ّ دي ّ شكهٓا Algorithm ٗطية ن ٕ زثيُا َطثق انكالو د ِ ػه

. ٖػايم اصا

Ex:

Sum (N)

T=0 ----------------------------------------->"Statement" (1)

(2)For (i1 to N) ---------------------------> "Loop” consist of (condition, statements)

(3) TT + (iii) ------------------------> "Statement”...But it's occurs N times.

(4) Return T----------------------------------->"Statement" Cost = 1

Analysis

(1) Cost = 1

(2) Condition Cost =N+

Assign Cost = 1

Incrementing = N

So Total = 2N+

(3) Body of the For Loop

Cost = 4 N (2 multi statments+1 sum statement + 1 assign statement)

Nٗيشات فا ُْعشب ف N ػهشا ٌ انه ٗ تيسصم انه ٗ ْٕا تي ٍ اقٕاط د ِ تيتكشس

Asymptotic Analysis (Notations)

Big O Notation:

َّد ِ ان ٗ اعتسان ّ ا algorithm ياػ ُٗ اقص ٗ زاخ ّ نم Upper boundٍان ٗ تيؼثش ػ Notation ال ٘د ..!ْيا نيٓا قا ٌَٕ كذ ِ يًك ٍ َثص ػهي .. !! ّ يؼذيٓا

𝑐𝑔(𝑛)د ِ ػًشٔا يا ْيؼذ ٖ ال algorithmَقذس َقٕل ا ٌ ال 𝑁 0 ا ٌ ي ٍ تؼذ ال ..ّيؼ ُٗ انشع ًّ دي

.. !!ممكن نقول مثال.. طبعا الكالم ده مش واضح خالص

Big O ىيبقى ال𝑇 𝑁 = 2 𝑁^2 بتاعتوا ىياtime complexityمعادلو الalgorithm لو مثال عندنا .. !!ُْٔشٕف االثثات د ِ دنٕقت ٗ نهكالو د ]ِ: 𝑂(𝑁^3 )بتعتوا ىيا notation

!!كم ان ٗ ُْؼًهٕا ُْؼٕض ف ٗ انقإٌَ

2 N^2 ≤ CN^3

2 ≤ CN

C = 1 and 𝑁 0 = 2

!! ؟𝑂(𝑁^2 ( طية ني ّ يش َقٕل

سظٕا َدشب تانقا ٌَٕ ب

2 𝑁^2 ≤ 𝐶𝑁^2

2 ≤ C

𝐶 = 2 𝑎𝑛𝑑 𝑁 0 = 1

upper bound ٍتتؼثش ػ 𝑂(𝑁^2 (ٔ يفيش يشاكم ف ٗ كذ ِ ال ٌ تاسدٔا ال! ٔاظر خذا إَا يُفغ

Omega Notation (  ):

ان ٗ اتكهًُا ػهيٓا قثم best caseان ٗ ْٕا ف ٗ ال lower boundان ٗ تتؼثش ػ ٍ ال Notation دي ّ تق ٗ ال .. !!ِكذ

cg(n)ٍاكثش ي 𝐹 𝑛 الصو يك ٌٕ ال 𝑁 0 ٍاكثش ي 𝑁 ٔيؼ ُٗ انشع ًّ دي ّ إَا ا ٖ قي ًّ نم

Example: 5N^2 = (N)-

∃𝐶, 𝑁 0 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 ∶ 0 ≤ 𝐶𝑁 ≤ 5 𝑁^2

CN ≤ 5 𝑁^2

C = 5 and N 0 = 1

n = 2 C 1 = (^1 )

فا طثؼا خذَا انشقى zero ٍٔ طثؼا اكثش ي 2 ٍتثق ٗ اكثش ي Nػهشا ٌ ْٗ انًفشٔض ال N 0 =2 ُْا خذَا ال ّيسققٕا انًؼادن C 1 , C 2 , N 0 ٔتًا اَُا نقيُا 2 االكثش انه ٗ ْٕا

:ٗيثق 1 2 𝑁

2 =^ (N

Algorithms:

Insertion Sort Algorithm:

...Sorting Problemنسم ال Algorithmدِ

Idea: ان .. !! ٗيثال في ّ اسقاو يش يتشتث ّ ٔ ػايض استثٓا array ٖد ِ ا ٌ تثغاط ّ ْيك ٌٕ ػُذ Algorithm فكش ِ ال : ْٗيسصم كاالت

ٔ ْؼذ ٖ ػه ٗ كم ال Key ِد Elementدي ّ ُْٔغ ًٗ ال arrayف ٗ ال Elementَْٗثيذأ ي ٍ تا key كم يش ِ تيثق ٗ يؼايا ...ٗد ِ تانشقى ان ٗ اَا ػُذٔا دنٕقت Key تؼذ كذ ِ ٔ كم يش ِ ْغيش ال Elements د ِ ْيغتًش طٕل يا اَا check ٔ ال .. !! ّان ٗ قثهي Elementsد ِ ػه ٗ ال keyتالcheck تفعم اػًم تؼذ كذ ِ تق ٗ تؼًم نالسقاو ان ٗ ْيا اكثش ي ٍ ال ...key ػهي ّ د ِ اكثش ي ٍال check تالق ٗ ا ٌ انشقى انه ٗ تؼًم ِكم يش array ٔ كذ ِ يثق ٗ ال key د ِ تق ٗ يكا ٌ اخش سقى نقيتٕا اكثش ي ٍ ال key ٔازط ال shift ّدي key نًا تذخم في ّ سقى خذيذ ف ٗ يكا ٌ انًُاعة ...shiftingٔ اندضء د ِ ْٕا ان ٗ تيسصهٕا sorted ٗخضء يُٓا تيثق Array ان ٗ يسافظ ػه ٗ تشتية ال

انًٕظٕع ْيًش ٗ اصاٖ؟ Insertion Sortتإعتخذاو الsortingّد ِ ٔػايضي ٍ َؼًه Arrayل ن ٕ افتشظُا ػُذَا ا

5 3 1 9 7

ٔازذ تظ Element َّاكيذ ال Sorted ِٔ د ِ ْؼتثش 5 انه ٗ ْٕا Element ْاخذ أل

ٖد 3 ٔ اشٕف ال 5 انه ٗ ْٕا ال Sorted ّْالق ٗ ٔ اتص ػه ٗ اندضء انه ٗ اَا قهت ػهي 3 تؼذ كذ ِ ْثص ػه ٗ ال يكآَا فيٍ؟

ٗنهثاق Shift فا تانتان ٗ ْسطٓا ف ٗ االٔل ٔ اػًم 5 ْالق ٗ آَا اقم ي ٍ ال

3 5 1 9 7

ٔ تؼذي ٍ اسخغ 1 َشٔذ ػه ٗ انه ٗ ػهي ّ انذٔس انه ٗ ْٕا ال Sortedتقٕا 2 Elements كذ ِ اَا اتأكذت ا ٌ أل انه ٗ ف ٗ االٔل Sorted ادٔس ػه ٗ يكا َّ ف ٗ اندضء ال

ٗنهثاق Shift ْالقي ّ اقم يثق ٗ ْسط ّ ف ٗ االٔل ٔ اػًم 3 ْقاس َّ تال

1 3 5 9 7

ٍػهشا ٌ اػشف يكآَا فيي Sorted ٗٔ اقاسَٓا تاندضء انه 9 تؼذي ٍ ْثص ػه ٗ ال

ٗتشظ ٕ يثق ٗ اسٔذ ػه ٗ انه 3 ْالقيٓا اكثش ي ُّ يثق ٗ ْشٔذ ػه ٗ انه ٗ تؼذي ّ ْالقي ّ اكثش ي ٍ ال 1 ْدشب ال ّفا ْسطٓا تؼذيٓا يؼ ُٗ ف ٗ انًثال د ِ يش ْغيش يكُٓا ْغيثٓا ْٗ كذ ِ يظثٕط 5 تؼذي ّ ْالقيٓا اكثش ي ٍ ال

انه ٗ قثهيٓا Sorted array ْشٔذ ادٔس ػه ٗ يكآَا ف ٗ ال 7 َشٕف ان ٗ تؼذيٓا ْالق ٗ ال

يثق ٗ ْسطٓا ف ٗ 9 ٔ ال 5 يثق ٗ ْٗ يكآَا تي ٍ ال 9 تظ يش اكثش ي ٍ ال 5 ٔ ال 3 ٔ ال 1 ْالقيٓا اكثش ي ٍ ال Shift يكُٓا ٔ اػًم

1 3 5 7 9 ِتظ كذ

Pseudo code: InsertionSort (Array, N) Array [1.......N]

For (i 2 to N)

KeyArray[i]

Ji-

While (j>=1 and key< Array[j])

Array [j+1]  Array[j]

J  j-

End While

Array [j+1] Key,

End For

Bubble sort Algorithm:

..Pseudo code طثؼا د ِ كهُا ػاسفي ُّ يش يستاج في ّ كالو تظ يًك ٍ َشٕف ال

Pseudo code: BubbleSort (A)

For i1 to length[A]

Do for j length[A] downto i+

Do if A[j] < A[j-1]

Then swap A[j]  A[j-1]

End for

Analysis: best caseٔ ف ٗ ال 𝑇 𝑛 = (𝑛^2 ) ْٕا worst caseد ِ ف ٗ ال algorithmتتاع الorder ال

.ٔ طثؼا انكالو د ِ ْيثق ٗ نًا االسقاو يتشتث ّ اعاعا 𝑇 𝑛 = (𝑁)ْيثقٗ

Done

By: A7med Mansour (Mans) && A7med Abdulla ( The Rock)

Introduction to Algorithm Analysis and Design: Lecture 1, Exams of Algorithms and Programming

Related documents

Partial preview of the text

Download Introduction to Algorithm Analysis and Design: Lecture 1 and more Exams Algorithms and Programming in PDF only on Docsity!

Lecture # 1

Introduction:

Calculating Run Time for any Algorithm:

Ex:

Analysis

Asymptotic Analysis (Notations)

Big O Notation:

Omega Notation (  ):

2 =^ (N

Algorithms:

Insertion Sort Algorithm:

Bubble sort Algorithm: