Complete Guide To Heaps

Heaps

A heap is a complete binary tree in which every node key is compared with its children and arranged accordingly. It is also called as a binary heap.
It has two properties :

• All levels are full, except the last one, which is filled from left to right
• Every node should have element greater (or smaller) than or equal to all its children nodes. (duplicates are allowed)

The ordering is of two types:

1. Max heap : A max heap is a complete binary tree in which the key value in each node is greater than or equal to the key values in the child node.

1. Min heap : A min heap is a complete binary tree in which the key value in each node is smaller than or equal to the key values in its child node.

   

As heap is a complete binary tree, height of the tree will be logN

Array Representation of Heaps

Heaps can be easily represented as an array since it is a complete binary tree.
Array-based representation is space-efficient.

Operations on Heap

Heapify

The process of creating a heap data structure froma binary tree is known as Heapify.
Min-Heap or a Max-Heap are constructed using this process.

1. create binary tree from array
2. Start from the first index of non-leaf node with index n/2-1.
3. set largest = i
4. index of :
   [left child] = 2i + 1
   [right child] = 2i + 2.
   If left child is greater than current node (i.e. element at ith index), set index of left child as largest.
   If right child is greater than current node , set index of right child as largest
5. Swap largest with current element.
6. Repeat steps 2-6 until the subtrees are also heapified.

void  Heapify(vector<int> &heap, int i)

{

int size = heap.size();

int largest = i;

int l = 2  * i + 1; //left child

int r = 2  * i + 2; //right child

if  (l < size && heap[l]  > heap[largest])

largest = l;

if  (r < size && heap[r]  > heap[largest])

largest = r;

if  (largest != i)

{

swap(&heap[i], &heap[largest]); // perform swapping

Heapify(heap, largest);

}

}

Insertion in Heap

Inserting in a Max heap

1. Create a new node. (If no node present)
2. If a node is already present, just insert the new node at the end (last node from left to right.)
3. Heapify

Note :For Min Heap, parent node is always smaller than newNode.

void  insert(vector<int> &heap, int val)

{

int size = heap.size();

if  (size == 0)

{

heap.push_back(val);

}

else

{

heap.push_back(val);

for  (int i = size/2-1; i >= 0; i--)

{

heapify(heap, i);

}

}

}

Deletion in Heap

To delete a node in Max heap:

1. Select value to be removed.
2. Swap it with the last element.
3. Remove the last element.
4. Heapify the tree.

Note : For Min Heap, both child nodes are greater than the current node.

void  hDelete(vector<int> &heap, int val)

{

int size = heap.size();

int i;

for  (i = 0; i < size; i++)

{

if  (val == heap[i])

break;

}

swap(&heap[i], &heap[size - 1]);

heap.pop_back();

for  (int i = size/2-1; i >= 0; i--)

{

heapify(heap, i);

}

}