Heapify

In the context of the heap data structure, heapify is an operation that ensures the heap property is maintained. The heap property states that for a max heap, for example, the parent node must be greater than or equal to its children.

Heapify is commonly used after an element is inserted or removed from a heap to rearrange the elements and maintain the heap property.

To perform heapify, we start from the last non-leaf node and compare it with its children. If the parent node is not greater than its children, we swap the parent node with the largest child. We then recursively perform heapify on the affected child subtree.

Here is an example of heapify in action using Java:

1class Main {
2  public static void main(String[] args) {
3    int[] arr = {9, 8, 7, 6, 5, 4, 3, 2, 1};
4    heapify(arr);
5
6    System.out.println("Heapified array: ");
7    for (int num : arr) {
8      System.out.print(num + " ");
9    }
10  }
11
12  public static void heapify(int[] arr) {
13    int n = arr.length;
14
15    for (int i = n / 2 - 1; i >= 0; i--) {
16      heapifyUtil(arr, n, i);
17    }
18  }
19
20  public static void heapifyUtil(int[] arr, int n, int i) {
21    int largest = i;
22    int left = 2 * i + 1;
23    int right = 2 * i + 2;
24
25    if (left < n && arr[left] > arr[largest]) {
26      largest = left;
27    }
28
29    if (right < n && arr[right] > arr[largest]) {
30      largest = right;
31    }
32
33    if (largest != i) {
34      int temp = arr[i];
35      arr[i] = arr[largest];
36      arr[largest] = temp;
37
38      heapifyUtil(arr, n, largest);
39    }
40  }
41}

In this example, we have an array arr containing elements in random order. We call the heapify function, which rearranges the elements to satisfy the heap property. The resulting heapified array is then printed.

Heapify is an important operation in heap data structures as it allows efficient insertion and deletion operations while maintaining the desired heap property.

xxxxxxxxxx

}

class Main {

  public static void main(String[] args) {

    // replace with your Java logic here

    int[] arr = {9, 8, 7, 6, 5, 4, 3, 2, 1};

    heapify(arr);

​

    System.out.println("Heapified array: ");

    for (int num : arr) {

      System.out.print(num + " ");

    }

  }

​

  public static void heapify(int[] arr) {

    int n = arr.length;

​

    for (int i = n / 2 - 1; i >= 0; i--) {

      heapifyUtil(arr, n, i);

    }

  }

​

  public static void heapifyUtil(int[] arr, int n, int i) {

    int largest = i;

    int left = 2 * i + 1;

    int right = 2 * i + 2;

​

    if (left < n && arr[left] > arr[largest]) {

      largest = left;

    }

​

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.

xxxxxxxxxx

}

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++;

  }

Min Heap

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

Similar to the max heap, we can represent a min heap in memory using 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 min heap property.

A min heap supports the following operations:

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

Here is an implementation of a min heap in Java:

1/* MinHeap class */
2```java
3
4class MinHeap {
5  private int[] heapArray;
6  private int maxSize;
7  private int heapSize;
8
9  public MinHeap(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 min heap. The maxSize variable represents the maximum size of the min heap, while heapSize represents the current number of elements in the min heap.

The insert(value) method inserts a new element into the min heap. If the min 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 min heap property.

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

The heapifyUp() method is used to fix the min 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 min heap property.

The heapifyDown() method is used to fix the min 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 min heap property.

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

xxxxxxxxxx

class MinHeap {

  private int[] heapArray;

  private int maxSize;

  private int heapSize;

​

  public MinHeap(int size) {

    maxSize = size;

    heapSize = 0;

    heapArray = new int[maxSize];

  }

​

  // Other methods...

}

Insertion in Heap

In the context of the heap data structure, insertion refers to the process of adding a new element to the heap while maintaining the heap property.

When inserting an element into a min heap, the new element is placed at the next available position in the heap array. The heap property is then restored by comparing the new element with its parent and swapping them if necessary until the heap property is satisfied.

Here is an example Java implementation of the insert(value) method in a MinHeap class:

1class MinHeap {
2  private int[] heapArray;
3  private int maxSize;
4  private int heapSize;
5
6  public MinHeap(int size) {
7    maxSize = size;
8    heapSize = 0;
9    heapArray = new int[maxSize];
10  }
11
12  public void insert(int value) {
13    if (isFull()) {
14      System.out.println("Heap is full. Cannot insert.");
15    } else {
16      heapArray[heapSize] = value;
17      heapSize++;
18      heapifyUp(heapSize - 1);
19    }
20  }
21
22  // Other methods...
23}

In the example code, we have a MinHeap class with an array heapArray to store the elements. The insert(value) method takes a value as input and checks if the heap is full. If the heap is not full, the new element is placed at the next available position in the array (heapSize) and heapifyUp() is called to restore the heap property.

The heapifyUp(index) method compares the element at the given index with its parent and swaps them if necessary to restore the heap property. This process continues recursively until the heap property is satisfied.

By performing the heapify up operation after inserting a new element, we can ensure that the heap property is maintained and the new element is properly positioned within the heap.

xxxxxxxxxx

class MinHeap {

  private int[] heapArray;

  private int maxSize;

  private int heapSize;

​

  public MinHeap(int size) {

    maxSize = size;

    heapSize = 0;

    heapArray = new int[maxSize];

  }

​

  public void insert(int value) {

    if (isFull()) {

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

    } else {

      heapArray[heapSize] = value;

      heapSize++;

      heapifyUp(heapSize - 1);

    }

  }

​

  // Other methods...

}

Deletion in Heap

Deleting an element from a heap involves removing an element and restoring the heap property.

In a max heap, the element with the highest value is always at the root of the heap. When we want to delete this element, we replace it with the last element in the heap and then perform a process called heapify down to restore the heap property.

Heapify down works by comparing the element at the root with its children and swapping it with the child that has the highest value. This process is repeated recursively until the heap property is satisfied.

Here is an example Java implementation of the deleteMax() method in a MaxHeap class:

1class MaxHeap {
2  private int[] heapArray;
3  private int heapSize;
4
5  public MaxHeap(int[] array) {
6    heapArray = array;
7    heapSize = array.length;
8    buildHeap();
9  }
10
11  public void deleteMax() {
12    if (isEmpty()) {
13      System.out.println("Heap is empty. Cannot delete.");
14    } else {
15      heapArray[0] = heapArray[heapSize - 1];
16      heapSize--;
17      heapifyDown(0);
18    }
19  }
20
21  // Other methods...
22}

In the example code, we have a MaxHeap class with an array heapArray to store the elements. The deleteMax() method checks if the heap is empty. If the heap is not empty, the root element (element with the highest value) is replaced with the last element in the array (heapArray[0] = heapArray[heapSize - 1]) and heapifyDown() is called to restore the heap property.

The heapifyDown(index) method compares the element at the given index with its children and swaps it with the child that has the highest value. This process continues recursively until the heap property is satisfied.

By performing the heapify down operation after deleting the root element, we can ensure that the heap property is maintained and the next highest element is at the root of the heap.