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)
  • Principles of algorithm analysis
  • Comparing growth rate of functions
  • Growth rate as a measure of efficiency of use of a faster processor
    
  
  
• Basic Algorithm Design Paradigms (CS 1510 Review)
  
• Topic: Greedy algorithms and exchange arguments
• Reading: Chapter 16
• 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
• Example: Knapsack (and Subset Sum)
  
• Example: Longest Increasing Subsequence
• Topic: Reductions and NP-completeness
• Reading: Chapter 34
• Example: 3SAT <= Vertex Cover
• Example: Vertex Cover <= Subset Sum
• Topic: Divide and conquer, Parallel Algorithms and recurrences
• Reading: Chapter 4, Chapter 27 and Chapter 30
• Example: Parallel Prefix on an EREW PRAM
  
• Divide and Conquer Recurrences
• Sums
  
• Solving recurrences using induction
• Example T(n) = 2 T(n/2) + 1

• Problem Complexity: Information Theoretic and Adversarial Lower Bounds
• Reading: Chapter 9
• Time complexity of a problem
• Example: Searching lower bound
• Example: Sorting lower bound
• Example: Element Uniqueness lower bound
  
• 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: Randomized Monte-Carlo Min-Cut Algorithm
• Las Vegas vs. Monte Carlo algorithms
• Bounding Tails and Union Bounds
  
• Example: Analysis of streaks in Bernoulli trials
• Example: High confidence analysis of Quicksort, Expected linear time selection, Randomly built Search trees
• binomaily distributed random variable and Chernoff bounds
• Yao's technique
• Example: Average case information theoretic lower bound for sorting
• Example: Randomized Las Vegas lower bound for sorting
  

• Topic: Graph Algorithms
• Graph Search Algorithms
• Reading: Chapters 22, 23, 24
• Example: Prim's minimum spanning tree algorithm
• Example: Dijkstra's shortest path algorithm
• Example: Shortest path in a directed acyclic graph, topological sorting, and depth first search
• Example: Bellman-Ford shortest path algorithm
  
• Network Flow
  
• Reading: Chapter 26
  
• Example: Ford-Fulkerson algorithm
• Example: Karp-Edmonds Algorithms
• Poly-time, pseudo poly-time, and strong poly-time
• Topic: Distributed Graph Algorithms
  
• Reading: Chapter 24 and 26
• Example: Bellman-Ford shortest path algorithm
• Example: Push Relabel network flow algorithm
  

• 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
• Weak duality, complementary slackness and statement of strong duality
  
• Example: Writing the diet problem as a linear program and computing its dual
  
• Example: Dual of network flow
• Example: Dual of the shortest path problem

• 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: Randomized 8/7 approximation for MAX3SAT
    
  • Example: Gap reduction for TSP without triangle inequality to prove non-approximatability 
  
  
• Topic: Online Algorithms
• Reading: Chapter 17
• Example: Lazy dictionary insertion
• Example: Multipop
• Example: Counting in Binary
• Example: Dynamic Table
  
• Example: Paging, general lower bound and resource augmentation analysis of LRU (paper)
• Example: Move to Front for List Update and general lower bound (paper1 paper2)
• Example: Analysis of Round Robin / Process Sharing and general lower bound for nonclairvoyant algorithms (paper)