UNIT – 1

1. Introduction to Algorithms

• Definition and Importance: Understanding what an algorithm is and its role in problem-solving.
• Characteristics of Algorithms: Correctness, efficiency, and simplicity.
• Algorithm Representation: Pseudocode, flowcharts.

UNIT – 2

2. Mathematical Foundations

• Mathematical Induction: Using induction to prove correctness and properties of algorithms.
• Summations and Recurrences: Solving summations and recurrence relations.
• Asymptotic Notations: Big O, Big Omega (Ω), Big Theta (Θ), little o, and little omega (ω).

UNIT – 3

3. Analysis of Algorithms

• Time Complexity: Best, average, and worst-case time complexity.
• Space Complexity: Measuring the space required by an algorithm.
• Amortized Analysis: Average time per operation over a worst-case sequence of operations.

UNIT – 4

4. Sorting and Searching Algorithms

• Basic Sorting Algorithms: Bubble sort, selection sort, insertion sort.
• Divide and Conquer Algorithms: Merge sort, quick sort, binary search.
• Linear Time Sorting: Counting sort, radix sort, bucket sort.

UNIT – 5

5. Divide and Conquer

• Concept and Methodology: Breaking down a problem into smaller subproblems, solving them independently, and combining their solutions.
• Examples: Binary search, merge sort, quick sort, Strassen’s matrix multiplication.

UNIT – 6

6. Dynamic Programming

• Concept and Methodology: Breaking down a problem into simpler subproblems, storing the results of subproblems to avoid redundant computations.
• Examples: Fibonacci numbers, knapsack problem, longest common subsequence, matrix chain multiplication.

UNIT – 7

7. Greedy Algorithms

• Concept and Methodology: Making the locally optimal choice at each stage with the hope of finding the global optimum.
• Examples: Huffman coding, Kruskal’s algorithm, Prim’s algorithm, Dijkstra’s algorithm.

UNIT – 8

8. Graph Algorithms

• Graph Representation: Adjacency matrix, adjacency list.
• Graph Traversals: Depth-first search (DFS), breadth-first search (BFS).
• Minimum Spanning Tree: Kruskal’s algorithm, Prim’s algorithm.
• Shortest Path Algorithms: Dijkstra’s algorithm, Bellman-Ford algorithm, Floyd-Warshall algorithm.

UNIT – 9

9. Backtracking and Branch and Bound

• Backtracking: Concept and methodology, solving problems by exploring all possible solutions and backtracking to find the optimal one.
• Examples: N-Queens problem, subset sum problem.
• Branch and Bound: Concept and methodology, solving optimization problems by systematically exploring candidate solutions.
• Examples: Travelling salesman problem, knapsack problem.

UNIT – 10

10. NP-Completeness

• Class P and NP: Definitions and significance.
• NP-Complete Problems: Understanding the concept, examples of NP-complete problems.
• Reducibility: Polynomial-time reductions between problems.

UNIT – 11

11. Advanced Topics

• Approximation Algorithms: Algorithms for NP-hard problems that find near-optimal solutions.
• Randomized Algorithms: Algorithms that use random numbers to influence their decisions.
• Parallel Algorithms: Algorithms designed to run on multiple processors simultaneously.

Recommended Books and Resources

• “Introduction to Algorithms” by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein: Comprehensive guide to algorithms and their analysis.
• “Algorithms” by Robert Sedgewick and Kevin Wayne: Detailed explanation of fundamental algorithms and data structures.
• “Algorithm Design” by Jon Kleinberg and Éva Tardos: Focuses on algorithm design techniques and their applications.
• Online Resources: MIT OpenCourseWare, Coursera, Khan Academy.

Practical Assignments

• Implementing Sorting Algorithms: Writing and analyzing different sorting algorithms.
• Dynamic Programming Problems: Solving classic dynamic programming problems.
• Graph Algorithms: Implementing and analyzing graph traversal and shortest path algorithms.
• NP-Complete Problems: Exploring solutions and reductions for NP-complete problems.
• Greedy Algorithms: Implementing and understanding greedy algorithms and their applications.

Practical Skills

• Algorithm Design: Developing efficient algorithms to solve complex problems.
• Algorithm Analysis: Analyzing the time and space complexity of algorithms.
• Problem-Solving: Applying algorithmic techniques to solve real-world problems.
• Coding Proficiency: Writing clean, efficient, and well-documented code.