Heap Data Structure

• always greater than its child node/s and the key of the root node is the largest among all other nodes. This property is also called max heap property.
• always smaller than the child node/s and the key of the root node is the smallest among all other nodes. This property is also called min heap property.

This type of data structure is also called a binary heap.

Heap Operations

Some of the important operations performed on a heap are described below along with their algorithms.

Heapify

Heapify is the process of creating a heap data structure from a binary tree. It is used to create a Min-Heap or a Max-Heap.

1. Let the input array be
2. Create a complete binary tree from the array
3. Start from the first index of non-leaf node whose index is given by n/2 - 1 .
4. Set current element i as largest .
5. The index of left child is given by 2i + 1 and the right child is given by 2i + 2 .
   If leftChild is greater than currentElement (i.e. element at ith index), set leftChildIndex as largest.
   If rightChild is greater than element in largest , set rightChildIndex as largest .
6. Swap largest with currentElement
7. Repeat steps 3-7 until the subtrees are also heapified.

Algorithm

Heapify(array, size, i) set i as largest leftChild = 2i + 1 rightChild = 2i + 2 if leftChild > array[largest] set leftChildIndex as largest if rightChild > array[largest] set rightChildIndex as largest swap array[i] and array[largest]

To create a Max-Heap:

MaxHeap(array, size) loop from the first index of non-leaf node down to zero call heapify

For Min-Heap, both leftChild and rightChild must be larger than the parent for all nodes.

Insert Element into Heap

Algorithm for insertion in Max Heap

If there is no node, create a newNode. else (a node is already present) insert the newNode at the end (last node from left to right.) heapify the array

1. Insert the new element at the end of the tree.
2. Heapify the tree.

For Min Heap, the above algorithm is modified so that parentNode is always smaller than newNode .

Delete Element from Heap

Algorithm for deletion in Max Heap

If nodeToBeDeleted is the leafNode remove the node Else swap nodeToBeDeleted with the lastLeafNode remove noteToBeDeleted heapify the array

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

For Min Heap, above algorithm is modified so that both childNodes are greater smaller than currentNode .

Peek (Find max/min)

Peek operation returns the maximum element from Max Heap or minimum element from Min Heap without deleting the node.

For both Max heap and Min Heap

return rootNode

Extract-Max/Min

Extract-Max returns the node with maximum value after removing it from a Max Heap whereas Extract-Min returns the node with minimum after removing it from Min Heap.

Python, Java, C/C++ Examples

# Max-Heap data structure in Python def heapify(arr, n, i): largest = i l = 2 * i + 1 r = 2 * i + 2 if l < n and arr[l] >arr[largest]: largest = l if r < n and arr[r] >arr[largest]: largest = r if largest != i: arr[i], arr[largest] = arr[largest], arr[i] heapify(arr, n, largest) def insert(array, newNum): array.append(newNum) current = len(array) - 1 while current > 0: parent = (current - 1) // 2 if array[current] > array[parent]: array[current], array[parent] = array[parent], array[current] current = parent else: break def deleteNode(array, num): size = len(array) i = 0 for i in range(size): if array[i] == num: break # Swap with the last element array[i], array[-1] = array[-1], array[i] array.pop() # Remove the last element which is now the number to be deleted # Only run heapify if the deleted node was not the last node if i < len(array): heapify(array, len(array), i) arr = [] insert(arr, 3) insert(arr, 4) insert(arr, 9) insert(arr, 5) insert(arr, 2) print("Max-Heap array:", arr) deleteNode(arr, 4) print("After deleting an element:", arr) 

// Max-Heap data structure in Java import java.util.ArrayList; class Heap < void heapify(ArrayListhT, int i) < int size = hT.size(); int largest = i; int l = 2 * i + 1; int r = 2 * i + 2; if (l < size && hT.get(l) >hT.get(largest)) largest = l; if (r < size && hT.get(r) >hT.get(largest)) largest = r; if (largest != i) < int temp = hT.get(largest); hT.set(largest, hT.get(i)); hT.set(i, temp); heapify(hT, largest); >> void insert(ArrayList hT, int newNum) < int size = hT.size(); if (size == 0) < hT.add(newNum); >else < hT.add(newNum); for (int i = size / 2 - 1; i >= 0; i--) < heapify(hT, i); >> > void deleteNode(ArrayList hT, int num) < int size = hT.size(); int i; for (i = 0; i < size; i++) < if (num == hT.get(i)) break; >int temp = hT.get(i); hT.set(i, hT.get(size-1)); hT.set(size-1, temp); hT.remove(size-1); for (int j = size / 2 - 1; j >= 0; j--) < heapify(hT, j); >> void printArray(ArrayList array, int size) < for (Integer i : array) < System.out.print(i + " "); >System.out.println(); > public static void main(String args[]) < ArrayListarray = new ArrayList(); int size = array.size(); Heap h = new Heap(); h.insert(array, 3); h.insert(array, 4); h.insert(array, 9); h.insert(array, 5); h.insert(array, 2); System.out.println("Max-Heap array: "); h.printArray(array, size); h.deleteNode(array, 4); System.out.println("After deleting an element: "); h.printArray(array, size); > >

// Max-Heap data structure in C #include int size = 0; void swap(int *a, int *b) < int temp = *a; *a = *b; *b = temp; >void heapify(int array[], int size, int i) < int largest = i; int l = 2 * i + 1; int r = 2 * i + 2; if (l < size && array[l] >array[largest]) largest = l; if (r < size && array[r] >array[largest]) largest = r; if (largest != i) < swap(&array[i], &array[largest]); heapify(array, size, largest); >> void insert(int array[], int newNum) < array[size] = newNum; size += 1; int current = size - 1; while (current != 0) < int parent = (current - 1) / 2; if (array[current] >array[parent]) < swap(&array[current], &array[parent]); current = parent; >else < break; >> > void deleteRoot(int array[], int num) < int i; for (i = 0; i < size; i++) < if (array[i] == num) break; >swap(&array[i], &array[size - 1]); // Reduce the size of the heap since the last element is now removed size -= 1; // Heapify from the current index to adjust the rest of the heap if (i < size) < heapify(array, size, i); >> void printArray(int array[], int size) < for (int i = 0; i < size; ++i) printf("%d ", array[i]); printf("\n"); >int main()

// Max-Heap data structure in C++ #include #include using namespace std; void swap(int *a, int *b) < int temp = *a; *a = *b; *b = temp; >void heapify(vector &hT, int i) < int size = hT.size(); int largest = i; int l = 2 * i + 1; int r = 2 * i + 2; if (l < size && hT[l] >hT[largest]) largest = l; if (r < size && hT[r] >hT[largest]) largest = r; if (largest != i) < swap(&hT[i], &hT[largest]); heapify(hT, largest); >> void insert(vector &hT, int newNum) < hT.push_back(newNum); int current = hT.size() - 1; // Bubble up while (current >0) < int parent = (current - 1) / 2; if (hT[current] >hT[parent]) < swap(&hT[current], &hT[parent]); current = parent; >else < break; >> > void deleteNode(vector &hT, int num) < int size = hT.size(); int i; for (i = 0; i < size; i++) < if (num == hT[i]) break; >swap(&hT[i], &hT[size - 1]); hT.pop_back(); // Update size after popping size = hT.size(); // Heapify from the current index to adjust the rest of the heap if (i < size) < heapify(hT, i); >> void printArray(const vector &hT) < for (int num : hT) cout int main() < vectorheapTree; insert(heapTree, 3); insert(heapTree, 4); insert(heapTree, 9); insert(heapTree, 5); insert(heapTree, 2); cout

Heap Data Structure Applications

• Heap is used while implementing a priority queue.
• Dijkstra's Algorithm
• Heap Sort

Table of Contents

• Definition
• Heapify
• Insert Element into Heap
• Delete Element from Heap
• Python, Java and C/C++ Examples
• Heap Data Structure Applications