Topics for the Midterm
CS 343: Analysis of Algorithms, Spring 2008

Please review class notes and chapters 1-4 (plus part of 5). Review written and programming homework.

Chapter 1: Stable Matching

• What is an algorithm?
• How do you solve the Stable Matching (and its variations)?
  • Proof of stability, perfection.
  • Implementation issues.
• General Lessons: How does one approach solving problems? Can one beat brute force?
• What is a graph, bipartite graph? Note how many problems can be cast as a problem in graph theory.
• Types of problems (degrees of hardness - will study more later)

Chapter 2: Complexity

• Asymptotic notation. Know the definitions of big-theta (Θ, tight), big-Oh (&Omicron), big-Omega (Ω), little-Omega (&omega), and little-Oh (&omicron). Note, this is really set notation!
• Know additivity and transitivity properties.
• When do you typically see
  • O(n log n) - divide and conquer
  • O(n²) - nested loops
  • O(n^k) - enumeration over subsets of size k
  • O(2ⁿ) - enumerate over all subsets.
• What does it mean to be poly-time? What does it mean for an algorithm to be efficient?
• Know relative behavior of common functions such as aⁿ, n^a, log(n), n!, etc
• How many ways are there of permuting n things?
• Know how to calculate the complexity of simple algorithms (e.g. loops, nested loops)
• Know the simple series: arithmetic (sum_k k), geometric (sum_k x^k), harmonic (sum_k 1/k)
• Know basic exponential and log identities, i.e. expand log(ab), log(a/b), log(a^b)
• Know how to manipulate summations.
• How do you do a complexity proof starting from the definition?
• What do limits (f/g) tell you about complexity?

Chapter 3: Graphs

• What is a graph. Directed vs undirected. Where are they used?
• Representation:
  • How are graphs represented: adjacency list and matrix.
  • How does representation affect complexity of an algorithm?
  • What are the pros and cons of each type of representation?
• Know graph terminology, e.g. path, simple, cycle, connected, tree, etc
• Traversal
  • How does one traverse a graph: DFS and BFS
  • What is complexity of each?
  • What is implementation?
• How does one find the shortest path between nodes?
• How does one determine of two nodes are connected? How does one identify all the connected components?
• How does one test if a graph is bipartite?
• Finding cycles.
• Directed vs undirected graphs.
• Directed Acyclic Graph (DAG)
• What is topological sorting? What is the algorithm?

Chapter 4: Greedy Algorithms

• What is a greedy algorithm?
• Applications:
  • Fractional knapsack problem (as opposed to 0-1 Knapsack problem)
  • Interval scheduling
  • Dijkstra's shortest path algorithm
  • Minimum spanning tree: Prim, Kruskal, Reverse-Delete, and the Union-find implementation.
  • Huffman coding
• How do you prove that a greedy algorithm gives an optimal solution:
  • Staying ahead: after each step of the greedy algorithm, its solution is at least as good as any other algorithm's.
  • Exchange Argument: Gradually transform any solution to the one found by the greedy algorithm without hurting its quality.

Chapter 5: Divide and Conquer (partial)

• Methods for evaluating complexity of recursive algorithms:
  • Unrolling/Telescoping.