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)
  • Comparing growth rate of functions
  • Growth rate as a mesure of efficiency of use of a faster processor
    
  • Sums
  
  
• Basic Algorithm Design Paradigms
  
• Topic: Greedy algorithms and exchange arguments
• Reading: Chapter 16
• Example: Max independent set in an interval graph
• Example: Maximal weight independent set in a matroid
  
• Example: Kruskal minimum spanning tree algorithm as an application of matroids
• 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: Minimum weight binary search tree
• Example: Longest Increasing Subsequence (problem 16.5.3)
• Example: Subset sum
• Example: Knapsack
• Topic: Divide and conquer, Parallel Algorithms and recurrences
• Reading:
  
• Divide and Conquer for multiplying polynomials/numbers
  
• EREW PRAM
• Example: Divide and conquer algorithm for associative operator
• Example: Divide and Conquer for parallel prefix
  
• Example: Merging and Sorting
• Divide and Conquer Recurrences
• Solving recurrences using induction, example T(n) = 2 T(n/2) + 1
  
• Designing algorithms through data structures
• Example:

• Topic: Information Theoretic and Adversarial Lower Bounds
• Reading: Chapter 9
• Time complexity of a problem
• Example: Searching lower bound
• Example: Sorting lower bound
• Example: n-1 lower bound for find the maximum
• Example: 3n/2 lower bound for computing median (not in the book)
• Example: Impossiblity of reach distributed consensus with one faculty processor (wikepedia entry)
  

• 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: Bloom filters
• Example: Universal Hash Functions
• Example: Min-cut
• Example: Miller-Rabin primality algorithm
• Las Vegas vs. Monte Carlo algorithms
• Bounding Tails
• Example: Analysis of streaks in Bernoulli trials
• binomaily distributed random variable and Chernoff bounds
  
• Example: High confidence analysis of Quicksort, Expected linear time selection, Randomly built Search trees
• Yao's technique
• Example: Average case information theoretic lower bound for sorting
• Example: Randomized lower bound for sorting
  

• 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

• Topic: Max flow
• Reading: Chapter 26
• Example: Ford-Fulkerson algorithm
• Example: Karp-Edmonds Algorithms
• Poly-time, pseudo poly-time, and strong poly-time
• Example: Randomized Monte-Carlo Min-Cut Algorithm
• Example: Reduction of Maximum Cardinality Bipartite Matching to Network Flow
  

• Linear Programming 
• Example: Writing network flow as a linear program
• Example: Writing shortest path as a linear program
  
• Simplex greedy algorithm vs. divide and conquer interior point methods
• Duality using the Diet problem
• Duality of network flow
• Duality of the shortest path problem
  

• Topic: NP-completeness and Reductions
  • Reading: Chapter 34
  • Example: Matrix Multiplication <= Matrix Squaring
    
  • Example: CNFSAT <= 3SAT  (local replacement)
    
  • Example: 3SAT <= Vertex Cover (component design)
  • Example: Self Reducibility of Vertex Cover
    
  • Example: Vertex Cover <= Set Cover (restriction)
    
  • Example: 3SAT <= Subset Sum  (component design)
    
  
  
• Topic: Approximation algorithms
  • Reading: Chapter 35
  • Example: Vertex Cover using maximal matching lower bound and LP lower bound
    
  • Example: 2 and 3/2 approximations for Traveling Salesman
  • Example: Set Cover
  • Example: Gap reduction for TSP without triangle inequality to prove non-approximatability 
  
  
• Topic: Online Algorithms
  • Example: Multipop
  • Reading: Move to Front for List Update (paper1 paper2)
  • Reading: Introduction to online scheduling (paper)
  • Example: Speed Scaling for flow plus energy (paper)