Building Heap from Array
Given an array of N elements. The task is to build a Binary Heap from the given array. The heap can be either Max Heap or Min Heap.
Example:
Input: arr[] = {4, 10, 3, 5, 1}
Output: Corresponding Max-Heap:
10
/ \
5 3
/ \
4 1
Input: arr[] = {1, 3, 5, 4, 6, 13, 10, 9, 8, 15, 17}
Output: Corresponding Max-Heap:
17
/ \
15 13
/ \ / \
9 6 5 10
/ \ / \
4 8 3 1
Suppose the given input elements are: 4, 10, 3, 5, 1.
The corresponding complete binary tree for this array of elements [4, 10, 3, 5, 1] will be:
4
/ \
10 3
/ \
5 1
Note:
Root is at index 0 in array.
Left child of i-th node is at (2*i + 1)th index.
Right child of i-th node is at (2*i + 2)th index.
Parent of i-th node is at (i-1)/2 index.
Simple Approach: Suppose, we need to build a Max-Heap from the above-given array elements. It can be clearly seen that the above complete binary tree formed does not follow the Heap property. So, the idea is to heapify the complete binary tree formed from the array in reverse level order following a top-down approach.
That is first heapify, the last node in level order traversal of the tree, then heapify the second last node and so on.
Time Complexity: Heapify a single node takes O(log N) time complexity where N is the total number of Nodes. Therefore, building the entire Heap will take N heapify operations and the total time complexity will be O(N*logN).
In reality, building a heap takes O(n) time depending on the implementation which can be seen here.
Optimized Approach: The above approach can be optimized by observing the fact that the leaf nodes need not to be heapified as they already follow the heap property. Also, the array representation of the complete binary tree contains the level order traversal of the tree.
So the idea is to find the position of the last non-leaf node and perform the heapify operation of each non-leaf node in reverse level order.
Last non-leaf node = parent of last-node.
or, Last non-leaf node = parent of node at (n-1)th index.
or, Last non-leaf node = Node at index ((n-1) - 1)/2.
= (n/2) - 1.
Illustration:
Array = {1, 3, 5, 4, 6, 13, 10, 9, 8, 15, 17}
Corresponding Complete Binary Tree is:
1
/ \
3 5
/ \ / \
4 6 13 10
/ \ / \
9 8 15 17
The task to build a Max-Heap from above array.
Total Nodes = 11.
Last Non-leaf node index = (11/2) - 1 = 4.
Therefore, last non-leaf node = 6.
To build the heap, heapify only the nodes:
[1, 3, 5, 4, 6] in reverse order.
Heapify 6: Swap 6 and 17.
1
/ \
3 5
/ \ / \
4 17 13 10
/ \ / \
9 8 15 6
Heapify 4: Swap 4 and 9.
1
/ \
3 5
/ \ / \
9 17 13 10
/ \ / \
4 8 15 6
Heapify 5: Swap 13 and 5.
1
/ \
3 13
/ \ / \
9 17 5 10
/ \ / \
4 8 15 6
Heapify 3: First Swap 3 and 17, again swap 3 and 15.
1
/ \
17 13
/ \ / \
9 15 5 10
/ \ / \
4 8 3 6
Heapify 1: First Swap 1 and 17, again swap 1 and 15,
finally swap 1 and 6.
17
/ \
15 13
/ \ / \
9 6 5 10
/ \ / \
4 8 3 1
Implementation:
C++
// C++ program for building Heap from Array #include <iostream> using namespace std; // To heapify a subtree rooted with node i which is // an index in arr[]. N is size of heap void heapify(int arr[], int n, int i) { int largest = i; // Initialize largest as root int l = 2 * i + 1; // left = 2*i + 1 int r = 2 * i + 2; // right = 2*i + 2 // If left child is larger than root if (l < n && arr[l] > arr[largest]) largest = l; // If right child is larger than largest so far if (r < n && arr[r] > arr[largest]) largest = r; // If largest is not root if (largest != i) { swap(arr[i], arr[largest]); // Recursively heapify the affected sub-tree heapify(arr, n, largest); } } // Function to build a Max-Heap from the given array void buildHeap(int arr[], int n) { // Index of last non-leaf node int startIdx = (n / 2) - 1; // Perform reverse level order traversal // from last non-leaf node and heapify // each node for (int i = startIdx; i >= 0; i--) { heapify(arr, n, i); } } // A utility function to print the array // representation of Heap void printHeap(int arr[], int n) { cout << "Array representation of Heap is:\n"; for (int i = 0; i < n; ++i) cout << arr[i] << " "; cout << "\n"; } // Driver Code int main() { // Binary Tree Representation // of input array // 1 // / \ // 3 5 // / \ / \ // 4 6 13 10 // / \ / \ // 9 8 15 17 int arr[] = { 1, 3, 5, 4, 6, 13, 10, 9, 8, 15, 17 }; int n = sizeof(arr) / sizeof(arr[0]); buildHeap(arr, n); printHeap(arr, n); // Final Heap: // 17 // / \ // 15 13 // / \ / \ // 9 6 5 10 // / \ / \ // 4 8 3 1 return 0; } |
Java
// Java program for building Heap from Array public class BuildHeap { // To heapify a subtree rooted with node i which is // an index in arr[].Nn is size of heap static void heapify(int arr[], int n, int i) { int largest = i; // Initialize largest as root int l = 2 * i + 1; // left = 2*i + 1 int r = 2 * i + 2; // right = 2*i + 2 // If left child is larger than root if (l < n && arr[l] > arr[largest]) largest = l; // If right child is larger than largest so far if (r < n && arr[r] > arr[largest]) largest = r; // If largest is not root if (largest != i) { int swap = arr[i]; arr[i] = arr[largest]; arr[largest] = swap; // Recursively heapify the affected sub-tree heapify(arr, n, largest); } } // Function to build a Max-Heap from the Array static void buildHeap(int arr[], int n) { // Index of last non-leaf node int startIdx = (n / 2) - 1; // Perform reverse level order traversal // from last non-leaf node and heapify // each node for (int i = startIdx; i >= 0; i--) { heapify(arr, n, i); } } // A utility function to print the array // representation of Heap static void printHeap(int arr[], int n) { System.out.println("Array representation of Heap is:"); for (int i = 0; i < n; ++i) System.out.print(arr[i] + " "); System.out.println(); } // Driver Code public static void main(String args[]) { // Binary Tree Representation // of input array // 1 // / \ // 3 5 // / \ / \ // 4 6 13 10 // / \ / \ // 9 8 15 17 int arr[] = { 1, 3, 5, 4, 6, 13, 10, 9, 8, 15, 17 }; int n = arr.length; buildHeap(arr, n); printHeap(arr, n); } } |
Python3
# Python3 program for building Heap from Array # To heapify a subtree rooted with node i # which is an index in arr[]. N is size of heap def heapify(arr, n, i): largest = i; # Initialize largest as root l = 2 * i + 1; # left = 2*i + 1 r = 2 * i + 2; # right = 2*i + 2 # If left child is larger than root if l < n and arr[l] > arr[largest]: largest = l; # If right child is larger than largest so far if r < n and arr[r] > arr[largest]: largest = r; # If largest is not root if largest != i: arr[i], arr[largest] = arr[largest], arr[i]; # Recursively heapify the affected sub-tree heapify(arr, n, largest); # Function to build a Max-Heap from the given array def buildHeap(arr, n): # Index of last non-leaf node startIdx = n // 2 - 1; # Perform reverse level order traversal # from last non-leaf node and heapify # each node for i in range(startIdx, -1, -1): heapify(arr, n, i); # A utility function to print the array # representation of Heap def printHeap(arr, n): print("Array representation of Heap is:"); for i in range(n): print(arr[i], end = " "); print(); # Driver Code if __name__ == '__main__': # Binary Tree Representation # of input array # 1 # / \ # 3 5 # / \ / \ # 4 6 13 10 # / \ / \ # 9 8 15 17 arr = [ 1, 3, 5, 4, 6, 13, 10, 9, 8, 15, 17 ]; n = len(arr); buildHeap(arr, n); printHeap(arr, n); # Final Heap: # 17 # / \ # 15 13 # / \ / \ # 9 6 5 10 # / \ / \ # 4 8 3 1 # This code is contributed by Princi Singh |
C#
// C# program for building Heap from Array using System; public class BuildHeap { // To heapify a subtree rooted with node i which is // an index in arr[].Nn is size of heap static void heapify(int []arr, int n, int i) { int largest = i; // Initialize largest as root int l = 2 * i + 1; // left = 2*i + 1 int r = 2 * i + 2; // right = 2*i + 2 // If left child is larger than root if (l < n && arr[l] > arr[largest]) largest = l; // If right child is larger than largest so far if (r < n && arr[r] > arr[largest]) largest = r; // If largest is not root if (largest != i) { int swap = arr[i]; arr[i] = arr[largest]; arr[largest] = swap; // Recursively heapify the affected sub-tree heapify(arr, n, largest); } } // Function to build a Max-Heap from the Array static void buildHeap(int []arr, int n) { // Index of last non-leaf node int startIdx = (n / 2) - 1; // Perform reverse level order traversal // from last non-leaf node and heapify // each node for (int i = startIdx; i >= 0; i--) { heapify(arr, n, i); } } // A utility function to print the array // representation of Heap static void printHeap(int []arr, int n) { Console.WriteLine("Array representation of Heap is:"); for (int i = 0; i < n; ++i) Console.Write(arr[i] + " "); Console.WriteLine(); } // Driver Code public static void Main() { // Binary Tree Representation // of input array // 1 // / \ // 3 5 // / \ / \ // 4 6 13 10 // / \ / \ // 9 8 15 17 int []arr = { 1, 3, 5, 4, 6, 13, 10, 9, 8, 15, 17 }; int n = arr.Length; buildHeap(arr, n); printHeap(arr, n); } } // This code is contributed by Ryuga |
Output:
Array representation of Heap is: 17 15 13 9 6 5 10 4 8 3 1
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