Unit 1 chapter 1 Design and Analysis of Algorithms
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The document introduces algorithms, outlining their definition, design, and implementation steps while highlighting important problem types and fundamental data structures. It explains the concept of algorithms through examples like finding the greatest common divisor (gcd) and generating prime numbers. The document also touches on algorithm efficiency, correctness, and various problem-solving techniques.
Unit 1 chapter 1 Design and Analysis of Algorithms
- 1. The Design and Analysis of Algorithms by Anany Levitin
- 2. 2 CHAPTER 1: INTRODUCTIONCHAPTER 1: INTRODUCTION What is an Algorithm Steps in Designing and Implementing an Algorithm Important Problem Types Fundamental Data Structures
- 3. 3 ALGORITHM A sequence of unambiguous instructions for solving a problem, i.e. for obtaining the required output for any legitimate input in a finite amount of time
- 4. Notation of the Algorithm 4
- 5. 5 Special Features of Algorithms Non-ambiguity Range of inputs The same algorithm can be represented in different ways Several algorithms for solving the same problem Different algorithms with different speeds
- 6. Find the GCD(m,n) Method 1: Descriptive Algorithm 6
- 7. GCD using Euclid’s Method gcd(m,n)=gcd(n,m nod n) Find: 1. GCD(124,12) 2. GCD(160,15) 3. GCD(338,22) 4. GCD(150,8) 7
- 8. Find the GCD(m,n) Method 2: Descriptive Algorithm 8
- 9. Find the GCD(m,n) Method 2: Pseudocode Representation 9
- 10. Find the set of all Prime numbers Method 1: Middle School Level Find Set of ALL Prime Numbers within 25 10
- 11. Find the set of all Prime numbers Method 2: Sieve of Erathosthenes Find Set of ALL Prime Numbers within 25 11
- 12. Fundamentals of Algorithmic Problem Solving / Algorithm Design & Analysis Process 12
- 13. 13 What does it mean to understand the problem? What are the problem objects? What are the operations applied to the objects? Deciding on computational means How the objects would be represented? How the operations would be implemented?
- 14. 14 Design an algorithm • Build a computational model of the solving process Prove correctness • Correct output for every legitimate input in finite time • Based on correct math formula • By induction
- 15. 15 Analyze the algorithm Efficiency: time and space Simplicity: Generality: range of inputs, special cases Optimality: no other algorithm can do better Coding How the objects and operations in the algorithm are represented in the chosen programming language?
- 16. 16 Important problem types Sorting Searching String processing Graph problems Combinatorial problems Geometric problems Numerical problems
- 17. 17 Fundamental data structures Linear data structures Array Linked list Stack Queue Operations: search, delete, insert Implementation: static, dynamic
- 18. 18 Non-linear data structures Graphs Trees : connected graph without cycles Rooted trees Ordered trees Binary trees Graph representation: adjacency lists, adjacency matrix Tree representation: as graphs; binary nodes Fundamental data structures
- 19. 19 Fundamental data structures Sets, Bags, Dictionaries Set: unordered collection of distinct elements Operations: membership, union, intersection Representation: bit string; linear structure Bag: unordered collection, elements may be repeated Dictionary: a bag with operations search, add, delete
- 20. 20 Conclusion Algorithm classification By types of problems By design technique Design techniques a general approach to solving problems
- 21. Check if you are Clear with: Definition of Algorithm Notation of Algorithm Examples: GCD and Prime Fundamentals of Algorithms / Process of Design & Analysis of Algorithms Types of Problems Fundamental Data Structures 21
- 22. Have a GREAT Day..!!! 22