Lecture 03-20-17

Optimal Binary Search Trees

• Minimizing weight * depth
• There is a property that is going to speed-up the program
  • Future DP: do problems have this property or not?
    • Makes significant improvement on runtime
• Example
  • When people tell you who you look like

  • Person	Frequency
    1	10
    2	7
    3	5
    4	6
    5	9
    6	12

  • T_1-224, 17, 1 (trees of 2 elements)
  • T_1-337, 22, 2 (trees of 3 elements)
  • T_2-4
• Choose roots
• Subtrees are split into “consecutive sets of items”
• 2 element trees, root would be larger
• For any Tij Keeping track of:
  • Cost of the best search tree on that set of elements
  • Save breakpoint (root) in order to reconstruct tree
  • Weight
    • Is equivalent to the cost of the tree
• For sets of 3:
  • If 1 root, tree for 2,3
  • If 2 root, tree for 1,3
• In general
  • Tij which is k,
  • weight(root) + cost(left) + weight(left) + cost(right) + weight(right) for each choice of root
• N^2 number of entries
• Takes at most time proportional to N = O(N^3)

Property

• Adding an element on the right can not move the root to the left
• Adding an element on the left can not move the root to the right
• Example
  • Say a tree had root 3, with subtrees T12, T45
  • Then, adding an element 6, then only need to search 3-6
• Not true for matrix chain multiplication
  • Will be a homework problem
• What is the order notation improvement?
  • From O(N^3) -> O(N^2)
• #times in row root increases: <= n
• #times in row root stays same: <= n
• Now O(n) per row now as opposed to O(n) per element

Common thought process:

• If I add an element to the right, can it move my root to the left?

HW4 Mergesort

• Worst thing to do: repeat a comparison
  • Wouldn’t split up the number of outcomes at all
• Heap sort will do that but mergesort will not
• Next worse thing:
  • Breaking it up so that only one of these outcomes is consistent with x < y
  • All others are consistent with x > y
  • Fundamentally, would like to split in 2
• For any value of k, before this comparison, there are k comparisons
  • Could have as many as k - 1 outcomes