Lab 08 - Heaps

Binary Heaps are binary trees with the following properties:

I. Any node's $ n$ value is greater then his children's $ c$ . ($ n.value \geq c.value$ )

II. The tree is almost complete (missing elements are possible only on the last level)

       15
     /    \
    9      5
   / \    / 
  2   3  1  

Heap representation using an array:

 15 9 5 2 3 1

This corresponds to a BF traversal of the tree, and it guarantees that the children nodes corresponding to v[i] are v[2i + 1] and v[2i + 2]

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1. Basic operations. Implement and analyse the complexity for the following operations:

1.1 empty_heap() - creates an empty heap.

1.2 insert(v, h) - inserts $ v$ in $ h$ . At the end of the procedure , $ h$ is the updated heap.

1.3 get_max(h) - return the maximum value from the heap.

1.4 delete(pos,h) - deletes element at position $ pos$ from the heap.

     algorithm:
        swap the last element with the element whose position is to be deleted
        the resulting tree may no longer be a heap
        (!!) make sure the heap property is preserved 

Deleting 9:

      15
    /    \
   9      5
  / \    /
 2   3  1  

9 is swapped with 1, then deleted

      15
    /    \
   1      5
  / \    /
 2   3  9  

Here, we make sure the heap property is preserved by swapping the largest of the children to be with the current root and repeating the process all over for the modified subtree:

      15
    /    \
   1      5
  / \ 
 2   3  

      15
    /    \
   3      5
  / \ 
 2   1 

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2. Using heaps:

2.1 Implement a procedure which constructs a heap from an arbitrary array, using the insert operation.

2.2 Implement heapsort using the previous procedure.