Variable Size Decrease

• Computing the Median
  • Clear O(n lg(n)) algorithm
    • Sort the array
    • Return the middle number.
  • How about the k^th number?
    • Sort the list, return it.
    • Partially sort the list to item k.
  • Both of these seem overkill.
  • A decrease and conquer algorithm would be 1/2 of quicksort
  • Partition
    • Divide the array around an element p, such that
    • All elements to the right of p are greater than or equal to p
    • All elements to the left of p are smaller than p.

    • LOMUTO_PARTITION(A,left, right)
      
       pivotElement ← A[left]
       split ← left
       for i ← left + 1 to right do
          if A[i] < pivotElement
              split ← split + 1
              swap(A[split],A[i])
       swap(A[left],A[split])
       return s
      
      	     

    • Trace for [ 5, 2, 6, 7, 4, 9, 3, 8, 10, 0]
    • In some sense, split is the candidate position for the partition.

    • QUICK_SELECT(A,left, right,k)
      
       if k < left or k > right 
           return ERROR 
       s ← LOMUTO_PARTITION(A,left,right)
       if s = k-1
          return A[s]
        else if  s > k
           return QUICK_SELECT(A, left,s-1)
        else
           return QUICK_SELECT(A, s+1,right)