Design and Analysis of Algorithms

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Course Description

Course Overview: Introduction to fundamental techniques for designing and analyzing algorithms, including asymptotic analysis; divide-and-conquer algorithms and recurrences; greedy algorithms; data structures; dynamic programming; graph algorithms; and randomized algorithms.

Required textbook: Kleinberg and Tardos, Algorithm Design, 2005. We will be covering most of Chapters 4–6, some parts of Chapter 13, and a couple of topics not in the book.

Prerequisites: Introduction to proofs, and discrete mathematics and probability (e.g., CS 103 and Stat116). If you have not taken a probability course, you should expect to do some independent reading during the course on topics in…

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Course Description

Course Overview: Introduction to fundamental techniques for designing and analyzing algorithms, including asymptotic analysis; divide-and-conquer algorithms and recurrences; greedy algorithms; data structures; dynamic programming; graph algorithms; and randomized algorithms.

Required textbook: Kleinberg and Tardos, Algorithm Design, 2005. We will be covering most of Chapters 4–6, some parts of Chapter 13, and a couple of topics not in the book.

Prerequisites: Introduction to proofs, and discrete mathematics and probability (e.g., CS 103 and Stat116). If you have not taken a probability course, you should expect to do some independent reading during the course on topics including random variables, expectation, conditioning, and basic combinatorics.


1. INTRODUCTION (1/4/2011)


  • Why are you here?
  • Example: Internet Routing
  • Shortest-Path Algorithms
  • Example: Sequence Alignment (Part 1)
  • Example: Sequence Alignment (Part 2)
  • Beating Brute Force Search
  • Administrivia
  • Recursive Algorithms for Integer Multiplication
  • Gauss's Trick

2. BASIC DIVIDE & CONQUER (1/6/2011)


  • Merge Sort: Motivation
  • Merge Sort: Formal Definition
  • Running Time of Merge
  • Running Time of Merge Sort (Part 1)
  • Running Time of Merge Sort (Part 2)
  • Guiding Principles of CS161 (Part 1)
  • Guiding Principles of CS161 (Part 2)
  • Review of Asymptotic Notation
  • Asymptotic Notation: Example #1
  • Asymptotic Notation: Example #2
  • Big-Omega and Big-Theta

3. THE MASTER METHOD (1/11/2011)


  • Integer Multiplication Revisited
  • Master Method: Formal Statement (Part 1)
  • Master Method: Formal Statement (Part 2)
  • Master Method: Examples
  • Proof of Master Method (Part 1)
  • Proof of Master Method (Part 2)
  • Master Method: Interpretation of the Three Cases
  • Proof of Master Method (Part 3)

4. LINEAR-TIME MEDIAN (1/13/2011) We apologize for the poor audio quality in this video.


  • The Selection Problem
  • Partitioning Around a Pivot
  • A Generic Selection Algorithm
  • Median of Medians
  • Recap
  • Rough Recurrence
  • Key Lemma (Part 1)
  • Key Lemma (Part 2)
  • The Substitution Method
  • Analysis of Rough Recurrence

5. GRAPH SEARCH & DIJKSTRA's ALGORITHM (1/18/2011)


  • Graph Primitives
  • Representing Graphs: Adjacency Matrices and Lists
  • Breadth-First and Depth-First Search
  • Dijkstra's Algorithm (Part 1)
  • Dijkstra's Algorithm (Part 2)
  • Dijkstra's Algorithm: Example
  • Dijkstra's Algorithm: Proof of Correctness (Part 1)
  • Dijkstra's Algorithm: Proof of Correctness (Part 2)
  • Undirected Connectivity

6. CONNECTIVITY IN DIRECTED GRAPHS (1/20/2011)


  • Strongly Connected Components
  • SCCs: A Two-Pass Algorithm
  • Depth-First Search Revisited
  • Example (Part 1)
  • Example (Part 2)
  • Two-Tier Structure of Directed Graphs
  • Correctness of Algorithm
  • Correctness Intuition
  • Proof of Key Lemma
  • Structure of the Web, Small World Property, and PageRank

7. Introduction to Greedy Algorithms (1/25/2011)


  • Course Roadmap
  • Application and Final Exam Info
  • A Scheduling Problem
  • Two Greedy Algorithms
  • Correctness Proof
  • Cost-Benefit Analysis

8. Minimum Spanning Trees (1/27/2011)


  • Introduction
  • Prim's Algorithm
  • Graph Theory Preliminaries
  • Feasibility of Prim's Algorithm
  • The Cut Property
  • Proof of Cut Property
  • Key Exchange Argument
  • Naive Running Time and Heap Review
  • Implementing Prim with Heaps (Part 1)
  • Implementing Prim with Heaps (Part 2)
  • New Running Time Analysis

9. Kruskal's Algorithm and Union-Find (2/1/2011)


  • Kruskal's Algorithm
  • Proof of Correctness (Part 1)
  • Proof of Correctness (Part 2)
  • Naive Running Time
  • Union-Find Data Structure
  • Union by Rank
  • Rank and Size of Subtrees
  • Open Research Question
  • Path Compression
  • Path Compression and the Ackermann Function

10. Path Compression and Clustering (2/3/2011)


  • Union-Find Review
  • Path Compression
  • Rank Blocks
  • Counting Pointer Updates
  • Clustering
  • A Greedy Algorithm
  • Correctness of Greedy Algorithm (Part 1)
  • Correctness of Greedy Algorithm (Part 2)

11. Introduction to Randomized Algorithms (2/8/2011)


  • The Min Cut Problem
  • The Contraction Algorithm
  • Probability Review
  • Analysis of Contraction Algorithm
  • Success Through Independent Trials
  • Final Comments

12. QuickSort (2/10/2011)


  • The QuickSort Algorithm
  • Best-Case and Worst-Case Pivots
  • Running Time of Randomized QuickSort
  • Probability Review Part 2
  • Linearity of Expectation
  • Counting Comparisons
  • Crux of Proof
  • Final Calculations
  • Lower Bound of Comaprison-Based Sorting

13. Hashing (2/15/2011)


  • Hashing: Introduction
  • Hashing: High-Level Idea
  • Running Time
  • How to Analyze Hashing
  • Universal Hashing
  • Proof of O(1) Running Time
  • A Universal Family
  • Universality: Proof Idea
  • Bloom Filters

14. Balanced Search Trees and Skip Lists (2/17/2011)


  • Review of Binary Search Trees
  • Deleting from a BST
  • Red-Black Trees
  • Height of Red-Black Trees
  • Rotations
  • Insertion to a Red-Black Tree
  • Skip Lists: High-Level Idea
  • Skip Lists: Intuition for Analysis

15. Introduction to Dynamic Programming (2/22/2011)


  • Dynamic Programming: A First Example
  • Structure of Optimal Solution
  • A Recursive Algorithm
  • Bottom-Up Formulation
  • Reconstruction Algorithm
  • The Knapsack Problem
  • Dynamic Programming Solution

16. Sequence Alignment (2/24/2011)


  • Sequence Alignment
  • Optimal Substructure
  • Dynamic Programming Solution
  • Dynamic Programming Algorithm
  • Shortest Paths with Negative Edge Lengths
  • On Negative Cycles
  • Optimal Substructure (Part 1)
  • Optimal Substructure (Part 2)

17. Shortest Paths: Bellman-Ford and Floyd-Warshall (3/1/2011)


  • Single-Source Shortest Paths Revisited
  • The Bellman-Ford Algorithm
  • Negative Cycle Checking
  • Space Optimization
  • The Floyd-Warshall Algorithm (Part 1)
  • The Floyd-Warshall Algorithm (Part 2)
  • Dynamic Programming Algorithm

18. NP-Complete Problems (3/3/2011)


  • Polynomial Time Algorithms and P
  • The Traveling Salesman Problem
  • Reductions
  • Completeness
  • NP-Completeness
  • Many Problems are NP-Complete
  • Does P=NP?
  • Coping with NP-Completeness
  • The Vertex Cover Problem
  • Smarter Brute-Force Search

19. Approximation Algorithms (3/8/2011)


  • Performance Guarantees for Heuristics
  • A Greedy Knapsack Algorithm
  • Proof of Performance Guarantee
  • Final Exam Info
  • Better Performance via Dynamic Programming
  • Accuracy Analysis
  • Running Time Analysis

20. The Wider World of Algorithms (3/10/2011)


  • Bipartite Matching
  • Stable Matching
  • Gale-Shapley Proposal Algorithm
  • Maximum Flow
  • Selfish Flow and Braess's Paradox
  • Linear Programming
  • Computational Geometry
  • Approximation and Randomized Algorithms
  • Complexity and Epilogue

Teacher: Prof. Tim Roughgarden

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