Introduction to Graduate Algorithms

This is a graduate-level course in the design and analysis of algorithms. We study techniques for the design of algorithms (such as dynamic programming) and algorithms for fundamental problems (such as fast Fourier transform or FFT).

In addition, we study computational intractability, specifically, the theory of NP-completeness. The main topics covered in the course include: dynamic programming; divide and conquer, including FFT; randomized algorithms, including RSA cryptosystem and hashing using Bloom filters; graph algorithms; max-flow algorithms; linear programming; and NP-completeness.

• Dynamic Programming
• Fibonacci Numbers, Longest Increasing Subsequence (LIS), Longest Common Subsequence (LCS),Knapsack, Chain Matrix Multiplication,Shortest Path Algorithms
• Randomized Algorithms
• Modular Arithmetic: Fast Modular Exponentiation, Multiplicative Inverses,RSA Cryptosystem: Fermat's Little Theorem, RSA Protocol, Primality Testing,Hashing: Traditional Chain Hashing, Bloom Filters
• Divide and Conquer
• Fast Integer Multiplication,Linear-Time Median,Fast Fourier Transform
• Graph Algorithms
• Strongly Connected Components, 2-Satisfiability,Minimum Spanning Tree,Markov Chains, PageRank
• Max-Flow Problems
• Ford-Fulkerson Algorithm,Max-Flow Min-Cut Theorem, Edmonds-Karp Algorithm,Max-Flow applied to Image Segmentation
• Linear Programming
• Simplex Algorithm,Weak and Strong Duality,Max-SAT Approximation
• NP-Completeness
• Complexity Classes: P, NP, NP-Complete,NP-Complete Problems: 3-SAT, Independent Set, Clique, Vertex Cover, Knapsack, Subset-Sum,Halting Problem