Max Heap

In the context of heap data structure, a max heap is a complete binary tree where the value of each node is greater than or equal to the values of its children. The root node of the max heap contains the maximum value in the heap.

To represent a max heap in memory, we typically use an array. The elements are inserted into the array from left to right, and each element is then arranged in such a way that it satisfies the max heap property.

A max heap supports the following operations:

• insert(value): Inserts a new element into the max heap while maintaining the max heap property.
• remove(): Removes and returns the maximum element from the max heap while maintaining the max heap property.
• isEmpty(): Returns true if the max heap is empty, false otherwise.
• isFull(): Returns true if the max heap is full, false otherwise.

Here is an implementation of a max heap in Java:

1// MaxHeap class
2```java
3
4class MaxHeap {
5  private int[] heapArray;
6  private int maxSize;
7  private int heapSize;
8
9  public MaxHeap(int size) {
10    maxSize = size;
11    heapSize = 0;
12    heapArray = new int[maxSize];
13  }
14
15  // Other methods...
16}

In this implementation, we use an array heapArray to store the elements in the max heap. The maxSize variable represents the maximum size of the max heap, while heapSize represents the current number of elements in the max heap.

The insert(value) method inserts a new element into the max heap. If the max heap is full, an error message is printed. Otherwise, the new element is inserted at the end of the array and then heapifyUp() is called to maintain the max heap property.

The remove() method removes and returns the maximum element from the max heap. If the max heap is empty, an error message is printed. Otherwise, the maximum element is replaced with the last element in the array, and heapifyDown() is called to maintain the max heap property.

The heapifyUp() method is used to fix the max heap property from the bottom up. It compares the element at the given index with its parent and swaps them if necessary to maintain the max heap property.

The heapifyDown() method is used to fix the max heap property from the top down. It compares the element at the given index with its children and swaps them if necessary to maintain the max heap property.

By using the underlying array representation and the heapifyUp() and heapifyDown() methods, we can efficiently insert and remove elements while maintaining the max heap property.

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}

class MaxHeap {

  private int[] heapArray;

  private int maxSize;

  private int heapSize;

​

  public MaxHeap(int size) {

    maxSize = size;

    heapSize = 0;

    heapArray = new int[maxSize];

  }

​

  public boolean isEmpty() {

    return heapSize == 0;

  }

​

  public boolean isFull() {

    return heapSize == maxSize;

  }

​

  public void insert(int value) {

    if (isFull()) {

      System.out.println("Heap is full. Cannot insert.");

      return;

    }

​

    heapArray[heapSize] = value;

    heapifyUp(heapSize);

    heapSize++;

  }