Analysis and Design of Algorithms

Course Information

  • Course Code: Not specified

  • Course Name: Analysis and Design of Algorithms

  • Level: 2nd Year / B.Sc.

  • Course Credit: 3 Credits

  • Instructor: Dr. Eman Monir

  • Teaching Assistant: Not specified

Course Material

  • Main Textbook: Cormen, Leiserson, Rivest, Stein, Introduction to Algorithms, 3rd Edition, 2009

  • Supplementary Materials: Slides available post-class

Table of Contents

  1. Foundations

    • Role of Algorithms

    • Design and Analysis

    • Growth of Functions

  2. Sorting

    • Heapsort

    • Quicksort

    • Linear Time Sorting

  3. Graph Algorithms

    • Elementary Graph Algorithms

    • Minimum Spanning Trees

  4. Advanced Techniques

    • Dynamic Programming

    • Greedy Algorithms

  5. Selected Topics

    • Linear Programming

    • String Matching

Algorithm Definition

  • An algorithm is a set of computational steps for solving a well-specified problem.

  • Key Properties:

    • Input: Values from a specified set.

    • Output: Solution values from a specified set.

    • Definiteness: Steps must be defined clearly.

    • Correctness: Produces correct outputs for all inputs.

    • Finiteness: Must terminate in a finite number of steps.

    • Effectiveness: Each step can be performed in finite time.

    • Generality: Applicable to all problems of the desired form.

Types of Problems Solved by Algorithms

  • Numerical Algorithms

  • Cryptographic Algorithms

  • Sorting and Searching Algorithms

  • Signal Processing Algorithms

  • Graph Algorithms

  • Computational Geometry Algorithms

  • Computer Vision and Graphics Algorithms

Algorithm Analysis and Design

  • Analysis: Predict algorithm cost in terms of resources and performance.

  • Design: Create efficient algorithms for problems using minimal time and space.

Complexity Concepts

  • Time Complexity: Function defining time required based on input size.

  • Space Complexity: Function defining memory used based on input size.

Algorithm Design Strategies

  • Brute Force

  • Divide and Conquer

  • Greedy Approach

  • Dynamic Programming

  • Backtracking

  • Space and Time Trade-offs

Sorting Problem

  • Problem statement: Sort a sequence in non-decreasing order.

  • Example of Sorting Algorithms:

    • Insertion Sort

    • Merge Sort

    • Selection Sort

Insertion Sort Analysis

  • Best Case: O(n) comparisons if already sorted.

  • Worst Case: O(n^2) comparisons if reversed sorted.

  • Average Case: O(n^2) comparisons for random sequences.

Themes of Algorithmic Discussions

  • Design efficiency

  • Expressing algorithms clearly

  • Proving correctness

  • Optimality in algorithms

Additional Algorithms Mentioned

  • Euclid’s Algorithm: For finding gcd(m,n).

  • Sieve of Eratosthenes: For listing primes up to n.

Conclusion

  • Algorithmic efficiency, design, and analysis are critical for successful computation tasks.