Unit 918: Advanced Algorithmic Paradigms and Techniques

1. Introduction to Algorithmic Paradigms

• Overview of algorithm design strategies
• Importance of problem decomposition and optimization

2. Divide and Conquer Algorithms

2.1 Concept and Principles

• Breaking problems into smaller subproblems
• Recursive problem solving
• Combining solutions

2.2 Classic Examples

• Merge Sort
• Quick Sort
• Binary Search

2.3 Analysis

• Recurrence relations
• Time complexity (Master theorem)

3. Greedy Algorithms

3.1 Fundamentals

• Greedy choice property
• Optimal substructure

3.2 Common Problems and Algorithms

• Activity Selection
• Huffman Coding
• Minimum Spanning Trees (Prim's and Kruskal's algorithms)

3.3 Correctness and Limitations

• When greedy algorithms work
• Counterexamples

4. Dynamic Programming

4.1 Principles

• Overlapping subproblems
• Memoization vs Tabulation

4.2 Classic Problems

• Fibonacci sequence
• Knapsack problem
• Longest Common Subsequence

4.3 Optimization Techniques

• Space optimization
• State reduction

5. Network Flow Algorithms

5.1 Introduction to Network Flows

• Definitions: flow networks, capacity, cuts

5.2 Ford-Fulkerson Method

• Augmenting paths
• Residual graphs

5.3 Edmonds-Karp Algorithm

• BFS for shortest augmenting paths
• Complexity analysis

5.4 Applications

• Bipartite matching
• Circulation with demands

6. Randomized Algorithms

6.1 Motivation and Types

• Randomization in algorithms
• Monte Carlo vs Las Vegas algorithms

6.2 Examples

• Randomized Quick Sort
• Randomized Min-Cut

6.3 Analysis

• Expected runtime
• Probability of correctness

7. Approximation Algorithms

7.1 NP-hard Problems and Motivation

• Why exact solutions are infeasible

7.2 Approximation Ratio and Performance Guarantees

7.3 Techniques

• Greedy approximation
• Local search

7.4 Example Problems

• Vertex Cover
• Traveling Salesman Problem (TSP) approximation

8. String Matching Algorithms

8.1 Problem Definition

• Pattern searching in text

8.2 Algorithms

• Knuth-Morris-Pratt (KMP) algorithm
• Boyer-Moore algorithm
• Rabin-Karp algorithm

8.3 Applications

• Text processing
• Bioinformatics
• Data retrieval

9. Computational Geometry Algorithms

9.1 Fundamentals

• Geometric representations
• Common problems

9.2 Algorithms

• Convex Hull (Graham Scan, Jarvis March)
• Line segment intersection
• Voronoi diagrams

9.3 Applications

• Computer graphics
• Geographic Information Systems (GIS)
• Robotics path planning

10. Parallel and Distributed Algorithms

10.1 Parallel Computing Models

• PRAM, message passing

10.2 Synchronization Techniques

• Locks, barriers

10.3 Load Balancing and Scalability

10.4 Example Algorithms

• Parallel prefix sum
• Distributed consensus

11. Online Algorithms

11.1 Definition and Challenges

• Incremental input processing
• Lack of full input knowledge

11.2 Competitive Analysis

• Measuring performance against offline algorithms

11.3 Techniques and Examples

• Online scheduling
• Caching algorithms (LRU, FIFO)

11.4 Regret Minimization

12. Summary and Integration

• Comparing paradigms
• Choosing the right approach
• Open problems and research directions