Lecture 03-24-17

Take Home Exam 4 Review:

1. Expand
2. Use the same reasoning for the big 5 selection algorithm
   • Boxes are of size n/3 so recurrence is T(2n/3)
   • Runtime with big 3 becomes O(nlogn)
3. Sorting comparisons
   • Selection sort will repeat
   • Insertion sort will not
4. Mergesort cannot repeat comparisons
   • At any point in time when comparisons are done in merges, you will output one of those to the final list
   • Show that mergesort does not do a good job of always dividing the number of comparisons in two
     • Reach [1, …, k] merge [k + 1, …, n]
     • Reach comparison last element k, with any position with [k + 1, …, n]
     • n orders of interweaving the last element with the other set
   • This leads to another algorithm that always come close to dividing possible outcomes in 2
     • Binary insertion sort
       • Why is this not the easiest nlogn sorting algorithm?
         • Have to shift elements
         • Sorting algorithms not always bounded on comparisons

Floyd’s Algorithm

• Advantage: negative weights without negative cycles
  • Will detect if Mii < 0
• Runtime: O(n³)
• How can we store n³ paths with only n² paths?
• Time to reconstruct the path, will take time proportional to length of path

NP completeness

• Trying to prove a problem is hard
• Resources
  • Gary & Johnson - Computers and Intractability
• Going to first limit ourselves to decision problems
  • Yes or no
• Optimization problems
  • Finding the best (optima BST, MST)
• Can have decision version of any of the optimization problems
• Showing that the decision problem is hard, then will show the optimization problem is hard as well
  • Optimization problem is at least as hard as decision

Classes of problems

• P: decision problems that can be solved in O(n^k)
• NP: non-deterministic polynomial
  • decision problems that have a polynomial length proof that the answer is yes
  • flipping decision does not mean that the inverse of the problem is in NP!
• Is P related to NP?
  • Suppose something is in P, is it in NP?
    • Yes
• P = NP?
  • Millenial problem
• Next time: prove a problem is difficult