Its a tree based data structure which is a complete binary tree(all nodes have 2 nodes, leaf-child may not have all 2), and satisfies a heap property. But, the amazing thing is that it uses array as its underlying data structure.

The heap is the implementation of the Priority Queue, where top most priority (or lowest priority) element is required to be at the top.

Heap Property

1. Max Heap

In this tree representation, every root node is always greater than both of its child. Note: right child can be lesser than left child.

Max Heap

2. Min Heap

As reverse of the max heap. Every root element is always smaller than both of its child elements.

Operations

Some basic operations on heap are:

  • heapify - To maintain either max or min heap property.
  • findMax or findMin - To get the top element of heap
  • insertElement - To insert a new element to the heap
  • extractMax or extractMin - Delete the top element, and return it
  • Get heapsize - Get the heap size
  • Increment/Decrement - For incrementing or decrementing priority or value of any node

How array represents the tree (Heap)

As I said earlier, they are based on array, not on any pointers stuff. Consider an array:

224,131,23,45,52,287,304,122,460,438

The first element will be the root of tree, 2nd and 3rd are the child of it. 4th and 5th will be child of 2nd node. And so on.

            224
           /    \
        131      23
       /   \    /   \
     45    52  287   304
    /  \    /
 122  460  438

Note: Above is just tree representation from array. It does not satisfy any max-heap or min-heap property.

Get Left child

leftChildIndex = rootIndex * 2 + 1

Get Right child

rightChildIndex = rootIndex * 2 + 2

Maintain Heap property - Max Heap

Lets assume we have to build a max heap. We would want every root node to be greater than its child nodes.

Maintain max heapify property for an index

void max_heapify(int[] arr, int index, int heapsize) {
    int l = 2 * index + 1;
    int r = 2 * index + 2;
    
    int indexOfLargest = index;
    if (l < heapsize && arr[l] > arr[index]) {
        indexOfLargest = l;
    }

    if (r < heapsize && arr[r] > arr[indexOfLargest]) {
        indexOfLargest = r;
    }
    
    if (indexOfLargest != index) {
        swap(arr, index, indexOfLargest);
        max_heapify(indexOfLargest);
    }
}

This operation takes O(log n)

Here for the index passed, we are comparing it with its child elements. If any child is greater than child, it will be swapped. Since, we have put a newer element to its kid. That sub-tree also has to satisfy max-heap property. And, we go in a natural recursive algorithm.

Build a max heap

void buildMaxHeap(int[] arr, int heapsize) {
    for (int i=heapsize/2; i>=0; i--) {
        max_heapify(arr, i, heapsize);
    }
}

This operation takes O(n log n) (including time complexity of max_heapify() as well)

Example

In above example of input, after buildMaxHeap, it will look like:

473,445,295,310,404,199,257,25,21,47

            473
           /    \
        445      295
       /   \    /   \
     310   404 199  257
    /  \    /
   25  21  47

Use cases of Heap Data Structure

Heap data structure has many usage, and is very useful data structure.

  • Used for implementing Priority Queue (Max/Min Priority Queue)
  • Maintaining n largest or smallest elements