Learning Objectives
4 objectives- Understand and apply key algorithmic paradigms including divide and conquer, greedy, and dynamic programming.
- Analyze and implement advanced algorithms for network flow, randomized decision-making, and approximation solutions.
- Explore specialized algorithmic areas such as string matching, computational geometry, parallel and distributed computing, and online algorithms.
- Develop skills to evaluate algorithm efficiency, correctness, and applicability across various problem domains.
Content Outline
PreviewUnit 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
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