Lecture 03-15-17

Discuss Exam

2. Detect cycles
   • a. Toplogical sort
   • b. Ways
     • DFS with back edges
     • Divide into connected components, see for each there are k-1 edges in component of k vertices
4. Cross edges
   • a. Undirected: 0
   • b. Directed: (n²)

Begin Dynamic Programming

• Divide into pieces and look from the top level
• Examples
  • factorial
  • fibonacci
    • Fib(20) >= 2 * fib(18) >= 2 * 2 * fib(16) = 2ⁿ
• Fundmental idea: save computed values to avoid recomputation
• Starting from the easiest
• DP gets correct solution for things that intuitive solutions are incorrect for

DP Problems

• Matrix chain multiplication
  • [i x j] times [j * k]
  • Standard multiplication takes O(i * j * k)
  • M1 * M2 * M3 … Mi
  • Some orders of multiplication takes fewer operations than others (since matrix multiplication is associative)
• Ideas:
  • Finding the largest and multiplying those first
    • Example of that doing badly:
      • [99 x 100] * [100 * 99] * [99 x 1]
        • (99 * 100 * 99) + (99 * 99 * 1) = ~10^6
        • (100 * 99 * 1) + (99 * 100 * 1) = ~10^4
  • Finding the smallest and multiplying those first:
    • Example of that doing badly
      • Didn’t show “maybe homework problem”
  • No one knows a greedy algorithm that solves this efficiently
  • Divide & conquer
    • n - 1 choices of how to divide
    • also exponential
• Approach:
  • How to bound the subproblems
  • Always multiplying consecutive matrices
  • “Easiest to hardest”