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Introduction to Graduate Algorithms

Georgia Institute of Technology via Udacity

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Overview

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.



Why Take This Course?

The design and analysis of algorithms form an essential basis for computer science. This course is useful for those who want to pursue advanced studies in computer science, as well as those who want to work as a software engineer.

Syllabus

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

Taught by

Arpan Chakraborty and Eric Vigoda

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