Tentative Course Outline

• Topic: Introduction to algorithm analysis and asymptotic notation
  • Reading: Chapters 1, 2 and 3 of CLR (Read the appendix covering mathematical background if you need to)
• Topic: Analysis of algorithms using sums
  • Reading: Chapter 6
  • Example: Building a heap, 7.3
• Topic: Divide and conquer, and recurrences
  • Reading: Chapter 5, Chapter 30, Chapter 9
  • Example: Divide and conquer algorithm for multiplying polynomials
  • Example: Fast Fourier transform (Skipping this semester)
  • Example: Linear time selection algorithm
  • Example: T(n)= T(n - sqrt{n}) + f(n)
• Topic: Information theoretic lower bounds
  • Reading: Chapter 9
  • Time complexity of a problem
  • Example: Searching lower bound
  • Example: Sorting lower bound
  • Example: Lower bound for element uniqueness (not in the book, skipping this semester)
  • Example: n-1 lower bound for find the maximum
  • Example: 3n/2 lower bound for computing median (not in the book)
• Topic: Average-case analysis, randomized algorithms, and bounding tails of distributions
  • Reading: Chapters 5, 7, 9, 11, 12, and 13, and Handout on randomized algorithms.
  • Linearity of expectations
  • Example: Hiring Problem
  • Example: Quicksort, Expected linear time selection, Randomly built Search trees
  • Example: Expected time for insert in closed addressed hash table
  • Example: Expected time for insert in open addressed hash table using uniform hashing
  • Example: Universal Hash Functions
  • Example: Analysis of streaks in Bernoulli trials
  • Example: High confidence analysis of Quicksort, Expected linear time selection, Randomly built Search trees
  • Example: Miller-Rabin Primality Algorithm
  • Example: Randomized Monte-Carlo Min-Cut Algorithm (not in the book, skipping this semester)
  • Las Vegas vs. Monte Carlo algorithms
  • Example: Average case information theoretic lower bound for sorting
  • Example: Randomized lower bound for sorting using Yao's technique
• Topic: Greedy algorithms
  • Reading: Chapter 16
  • Example: Max independent set in an interval graph
  • Example: Coloring an interval graph (skipping this semester)
  • Example: Huffman Coding (skipping this semester)
  • Matroids
  • Example: Weighted unit tasks as an application of matroids (skipping this semester)
  • Example: Kruskal minimum spanning tree algorithm as an application of matroids
  • Recommended Problems: 16-1, 16-2
• Topic: Dynamic programming
  • Reading: Chapter 15
  • Suggested Reading: A paper on how to design dynamic programming algorithms for some problems without using recursion. (postscript)
  • Example: Fibonacci numbers
  • Example: Longest Common Subsequence
  • Example: Matrix Chain Multiplication (Skipping this semester)
  • Example: Minimum weight binary search tree
  • Example: Longest Increasing Subsequence (problem 16.5.3)
  • Example: Subset sum
  • Example: Knapsack
• Topic: Amortization
  • Reading: Chapters 17 and 21
  • Example: Multipop
  • Example: Incrementing Binary Counter
  • Example: Dynamic Table Expansion
  • Example: Union-Find (Chapter 21) (Skipping this semester)
  • Recommended Problems: 17.3-3, 17.4-3, 17-2, 17-3, 21.2-3, 21.3-4, 21-3
• Topic: Graph Search Algorithms
  • Reading: Chapters 22, 23, 24
  • Example: Prims Algorithm
  • Example: Dijkstra's Algorithm
  • Example: Bellman Ford
  • Example: Topological Sort/Shortest Paths in a DAG
  • Example: Depth first search (skipping this semester)
  • Example: Strongly connected components (skipping this semester)
• Topic: Max flow
  • Reading: Chapter 26
  • Example: Ford-Fulkerson algorithm
  • Example: Maximum Matching
  • Example: Karp-Edmonds Algorithms
  • Poly-time, pseudo poly-time, and strong poly-time
  • Example: Push-Relabel algorithms (probably skipping this semester)
• Linear Programming 
• Example: Writing network flow as a linear program
• Simplex greedy algorithm vs. divide and conquer interior point methods
• Duality
  
• Topic: NP-completeness and Reductions
  • Reading: Chapter 34
  • Example: sorting <= TSP
  • Example: CNFSAT <= SAT
  • Example: CNFSAT <= 3SAT
  • Example: 3SAT <= Vertex Cover
  • Example: 3SAT <= Subset Sum
  • Example: Gap reduction from Hamiltonian cycle to TSP
• Topic: Approximation algorithms
  • Reading: Chapter 35
  • Example: Vertex Cover
  • Example: Traveling Salesman
  • Example: Set Cover
  • Example: Turning dynamic programs into PTAS for subset selection problems (reading)  (May be skipping this semester)
  • Example: Subset Sum as an example of the above technique (May be skipping this semester)
• Topic: Online Algorithms
  • Reading: Introduction to competitive analysis (paper1 paper2)
  • Reading: Introduction to online scheduling (paper)
  • Reading: Speed  scaling for weighted flow (paper)