A max-heap is a complete binary tree in which the value in each internal node is greater than or equal to the values in the children of that node. Mapping the elements of a heap into an array is trivial: if a node is stored an index k, then its left child is stored at index 2k + 1 and its right child at index 2k + 2.
Illustration: Max Heap
How is Max Heap represented?
A-Max Heap is a Complete Binary Tree. A-Max heap is typically represented as an array. The root element will be at Arr[0]. Below table shows indexes of other nodes for the ith node, i.e., Arr[i]:
Arr[(i-1)/2] Returns the parent node.
Arr[(2*i)+1] Returns the left child node.
Arr[(2*i)+2] Returns the right child node.
Operations on Max Heap are as follows:
- getMax(): It returns the root element of Max Heap. The Time Complexity of this operation is O(1).
- extractMax(): Removes the maximum element from MaxHeap. The Time Complexity of this Operation is O(Log n) as this operation needs to maintain the heap property by calling the heapify() method after removing the root.
- insert(): Inserting a new key takes O(Log n) time. We add a new key at the end of the tree. If the new key is smaller than its parent, then we don't need to do anything. Otherwise, we need to traverse up to fix the violated heap property.
Note: In the below implementation, we do indexing from index 1 to simplify the implementation.
Methods:
There are 2 methods by which we can achieve the goal as listed:
- Basic approach by creating maxHeapify() method
- Using Collections.reverseOrder() method via library Functions
Method 1: Basic approach by creating maxHeapify() method
We will be creating a method assuming that the left and right subtrees are already heapified, we only need to fix the root.
Example
Java // Java program to implement Max Heap // Main class public class MaxHeap { private int[] Heap; private int size; private int maxsize; // Constructor to initialize an // empty max heap with given maximum // capacity public MaxHeap(int maxsize) { // This keyword refers to current instance itself this.maxsize = maxsize; this.size = 0; Heap = new int[this.maxsize]; } // Method 1 // Returning position of parent private int parent(int pos) { return (pos - 1) / 2; } // Method 2 // Returning left children private int leftChild(int pos) { return (2 * pos) + 1; } // Method 3 // Returning right children private int rightChild(int pos) { return (2 * pos) + 2; } // Method 4 // Returning true if given node is leaf private boolean isLeaf(int pos) { if (pos > (size / 2) && pos <= size) { return true; } return false; } // Method 5 // Swapping nodes private void swap(int fpos, int spos) { int tmp; tmp = Heap[fpos]; Heap[fpos] = Heap[spos]; Heap[spos] = tmp; } // Method 6 // Recursive function to max heapify given subtree private void maxHeapify(int pos) { if (isLeaf(pos)) return; if (Heap[pos] < Heap[leftChild(pos)] || Heap[pos] < Heap[rightChild(pos)]) { if (Heap[leftChild(pos)] > Heap[rightChild(pos)]) { swap(pos, leftChild(pos)); maxHeapify(leftChild(pos)); } else { swap(pos, rightChild(pos)); maxHeapify(rightChild(pos)); } } } // Method 7 // Inserts a new element to max heap public void insert(int element) { Heap[size] = element; // Traverse up and fix violated property int current = size; while (Heap[current] > Heap[parent(current)]) { swap(current, parent(current)); current = parent(current); } size++; } // Method 8 // To display heap public void print() { for (int i = 0; i < size / 2; i++) { System.out.print("Parent Node : " + Heap[i]); if (leftChild(i) < size) // if the child is out of the bound // of the array System.out.print(" Left Child Node: " + Heap[leftChild(i)]); if (rightChild(i) < size) // the right child index must not // be out of the index of the array System.out.print(" Right Child Node: " + Heap[rightChild(i)]); System.out.println(); // for new line } } // Method 9 // Remove an element from max heap public int extractMax() { int popped = Heap[0]; Heap[0] = Heap[--size]; maxHeapify(0); return popped; } // Method 10 // main driver method public static void main(String[] arg) { // Display message for better readability System.out.println("The Max Heap is "); MaxHeap maxHeap = new MaxHeap(15); // Inserting nodes // Custom inputs maxHeap.insert(5); maxHeap.insert(3); maxHeap.insert(17); maxHeap.insert(10); maxHeap.insert(84); maxHeap.insert(19); maxHeap.insert(6); maxHeap.insert(22); maxHeap.insert(9); // Calling maxHeap() as defined above maxHeap.print(); // Print and display the maximum value in heap System.out.println("The max val is " + maxHeap.extractMax()); } } OutputThe Max Heap is Parent Node : 84 Left Child Node: 22 Right Child Node: 19 Parent Node : 22 Left Child Node: 17 Right Child Node: 10 Parent Node : 19 Left Child Node: 5 Right Child Node: 6 Parent Node : 17 Left Child Node: 3 Right Child Node: 9 The max val is 84
Method 2: Using Collections.reverseOrder() method via library Functions
We use PriorityQueue class to implement Heaps in Java. By default Min Heap is implemented by this class. To implement Max Heap, we use Collections.reverseOrder() method.
Example
Java // Java program to demonstrate working // of PriorityQueue as a Max Heap // Using Collections.reverseOrder() method // Importing all utility classes import java.util.*; // Main class class GFG { // Main driver method public static void main(String args[]) { // Creating empty priority queue PriorityQueue<Integer> pQueue = new PriorityQueue<Integer>( Collections.reverseOrder()); // Adding items to our priority queue // using add() method pQueue.add(10); pQueue.add(30); pQueue.add(20); pQueue.add(400); // Printing the most priority element System.out.println("Head value using peek function:" + pQueue.peek()); // Printing all elements System.out.println("The queue elements:"); Iterator itr = pQueue.iterator(); while (itr.hasNext()) System.out.println(itr.next()); // Removing the top priority element (or head) and // printing the modified pQueue using poll() pQueue.poll(); System.out.println("After removing an element " + "with poll function:"); Iterator<Integer> itr2 = pQueue.iterator(); while (itr2.hasNext()) System.out.println(itr2.next()); // Removing 30 using remove() method pQueue.remove(30); System.out.println("after removing 30 with" + " remove function:"); Iterator<Integer> itr3 = pQueue.iterator(); while (itr3.hasNext()) System.out.println(itr3.next()); // Check if an element is present using contains() boolean b = pQueue.contains(20); System.out.println("Priority queue contains 20 " + "or not?: " + b); // Getting objects from the queue using toArray() // in an array and print the array Object[] arr = pQueue.toArray(); System.out.println("Value in array: "); for (int i = 0; i < arr.length; i++) System.out.println("Value: " + arr[i].toString()); } } OutputHead value using peek function:400 The queue elements: 400 30 20 10 After removing an element with poll function: 30 10 20 after removing 30 with remove function: 20 10 Priority queue contains 20 or not?: true Value in array: Value: 20 Value: 10