Heap Sort

Objectives

Notes

• Remeber a heap is a structure with expected:
  • Insertion time of O(log₂ n)
  • Deletion time of O(log₂ n)
• A max-heap keeps the largest value at the top of the heap.
• A heap is implmented as a nearly complete binary tree.
• A heap can be implmented in an array
  • parent(i) = return ⌊(i-1)/2⌋
  • Left(i) = return 2i +1
  • Right(i) = return 2i+2

• index	0	1	2	3	4	5	6
  Value	Root	L0	R0	L1	R1	L2	R2

• The Sort

  HEAP-SORT (A)
  
   BUILD-MAX-HEAP(A)
   For i < A.length-1 downto 1
      swap(A[0], A[i]
      heap_size --; 
      MAX-HEAPIFY(A, 0)
  
  
  BUILD-MAX-HEAP(A)
  
   heap_size ← A.length-1
   for i ← ⌊ A.length/2 ⌋ down to 0
       MAX-HEAPIFY(A,i)
  
  
  MAX-HEAPIFY(A,i)
  
   l ← left(i)
   r ← righ(i)
   pos ← LargestPos(i, l)
   pos ← LargestPos(pos, r)
   If pos ≠ i
      swap(A[i], A[pos])
      MAX-HEAPIFY(A, pos)
  
  
  LargestPos (a,b)
  
   rv ← i
   if b < heap_size
      if(A[b] > A[i]
         rv ← b
   return rv
  
  

• Try MAX-HEAPIFY with {4, 1, 3, 2, 6, 5, 7}
• Try HEAP-SORT with {4, 1, 3, 2, 6, 5, 7}