✎ aa:lab:08 [books]

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====== Lab 08 - Heaps ======

==== Heaps ====

Binary Heaps are binary trees with the following properties:

 **I.**  Any node's $math[n] value is greater then his children's $math[c]. ($math[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]

----


**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 $math[v] in $math[h]. At the end of the procedure , $math[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 $math[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 

----

**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.