Lecture 03-13-17

Selection

• Find the kth largest from set of size n
  • Easy: sort and find, O(nlogn)
  • Find it in O(n) time
    • Suppose you can magically find the median in O(n) time
    • Then you can find the kth largest for any k
      • Why?
      • Divide into less than median, equal to median, greater than median
      • Recurrence: T(n) <= cn + T(n/2)
        • Easy to solve: 2cn so O(n)
        • cn is to find the median
      • Say instead of finding median, found close to median
        • Approximation is close to n/3 and 2n/3
        • T(n) <= cn + T(2n/3)
          • Still O(n)
      • Goal now: find an element that is provably close to the median

Finding element that is provably close to the median

• Example:
  • 25 numbers
  • Groups of 5
• First take the median of each 5 element group: O(n) overall
• Recursively find the true median of the set (these medians)
  • T(n) = T(n/5)
  • Not necessarily true median of all the values
  • Argue that the median of these sets is close to the median of the whole set
• Steps:
  1. Find median of 5 element groups
  2. Recursively find the median of these medians: m
  3. Divide groups into
     • s1 < m
     • s2 = m
     • s3 > m
  4. Recursively search for value in s1 or s3

Runtime analysis

• As an exercise: sort groups and find true median
• If sort, top left of grid is < m, bottom right is > m
• For top right and bottom left, can’t guarantee anything
  • No matter what happens, one of these boxes is eliminated
  • Worst: eliminate one of these boxes and nothing else
    • Question: what is the size of the boxes?
    • Recall: #rows will usually be much > #columns
    • Eliminating 3 elements from n/5 rows but only 1/2 of these
    • Necessarily eliminating 3n/10 elements
    • Largest that the recursive call can be: 7n/10 (n - 3n/10)
    • T(n) <= cn + T(n/5) + T(7n/10)
      • T(n/5) is taking the recursively taking medians and finding median
• Analysis of steps 1. 2. 3.
  1. Recurrence depends on how close the calculated median was to the true median
     • How close to the median can I guarantee the median of the medians is
• Some people will call this a divide & conquer algorithm, some will not

Runtime analysis of T(n) <= cn + T(n/5) + T(7n/10)

1. Induction
   • Instead proving T(n) <= 10cn
   • Basis: “really not a problem”
   • Inductive hypothesis
     • for all i < n, T(i) <= 10ci
   • Inductive step:
     • T(n) <= cn + 2cn + 7cn = 10cn
2. Technique: Guess & prove
   • Think you should see it once
   • Figure out: I think it’s O(n)
   • cn + k(n / 5) + k(7n/10) <= kn
   • c + k/5 + k7/10 <= more math
   • c <= k/10
   • Show wrong guess example with mergesort

The Big-5 Algorithm