A binary tree is a tree in which a node can have a maximum of two offspring, while a heap is a full binary tree. A full binary tree has all levels filled in, except for the last level, or leaf node, with all nodes aligned to the left.
The Heapsort Algorithm will be discussed in this post. Heap sort is responsible for processing the items by utilising the components of the provided array to create the max-heap or min-heap. The ordering of an array is known as a max-heap or min-heap, with the root member serving as the array's maximum or minimum element.
Let's first take a quick look at the description of heap sort before learning more about it.
| Table of Content |
What Is Heap Sort?
Heap sort refers to the algorithm which is a comparison-based in-place sorting method that uses the available memory space and doesn't require any more memory. As the name implies, heap sort uses a data structure called a heap to sort the supplied array. Due to its advantageous worst-case time complexity, it is one of the most favoured Sorting in Data Structures algorithms.
Like how selection sort separates an array into two parts, heap sort likewise does so.
- Sorted region
- Unsorted region
We have the unsorted area of the array at first, with the sorted region being empty. Iteratively, the biggest element is selected from the unsorted region and added to the sorted region. A special kind of full binary tree called a heap represents the elements in an unsorted area of the array.
What is Heap Data Structure?
Whether it is a max-heap or a min-heap, a heap is a full binary tree where the nodes are arranged differently.
Each given node in a heap data structure, which meets the heap property, is one of the following:
- The root node's key is the biggest of all the nodes and always greater than any of its child nodes. The max heap property is another name for this property.
- The root node's key is the smallest of all the nodes and always smaller than the child node or nodes. It is also known as the minimum heap property.
Another name for this kind of data structure is a binary heap.You can also check Business Analytics and Data Science program by Hero Vired.
How Does Heap Sort Algorithm Work?
There are 2 fundamental components to a heap sort algorithm in Data Structures. The unsorted list or array is first compiled into a heap, and the sorted array is then made by continually removing the largest and smallest elements from the heap and adding them to the array. With each elimination, the heap is rebuilt.
The initial heap sort stage generates a Heap data structure when given an unsorted list (Max-Heap or Min-Heap). When a heap is formed, the first member will be the largest or the smallest (depending on whether Max-Heap or Min-Heap is used). Thus we will place that element in our array. Finally, we use the remaining items to create a heap again, selecting the first piece and inserting it into the array. We continuously perform the same thing until we have the entire sorted list in our array.
How To "Heapify" A Tree?
We can convert a complete binary tree into a Max-Heap using the heapify function on all the non-leaf items in a heap. It might be challenging to understand heapify since it employs recursion. Heapify is the process of transforming a valid heap data structure from a binary tree.
The typical method is to start at the last internal node and work up to the root node, exchanging each node with the largest child node unless the heap property is met. The procedure is repeated for the impacted subtrees until the entire tree is a valid heap. Heapify has an O(log n) time complexity, where n refers to the number of nodes in the tree.
Heap Sort Complexity
The number of levels the new element must ascend to meet the heap property determines the number of operations needed. As a result, the worst-case time complexity of the insertion operation is O. (log n). The average-case complexity of the insertion operation is O for a random heap and repeated insertions (1).
Time Complexity
The length of the input determines how long it takes an heap sort algorithm in Data Structures to run, known as temporal complexity. It calculates how long each algorithm's code statement takes to run.
Best Case Complexity - When there’s no need for sorting, that is when the array is sorted already, it gives rise to the best-case complexity. The heap sort's best-case time complexity will be O (n logn).
Average Case Complexity - This happens when the array's items are arranged in an inconsistent manner that is neither correctly ascending nor downward. The heap sort's average case time complexity is O (n log n).
Worst Case Complexity- It happens when the array's items are sorted in a reversible manner. The array's items are in descending order. Therefore, let's say you need to sort them in ascending order. The heap sort's worst-case time complexity is O (n log n).
The heap sort's time complexity in all three scenarios is O (n logn). A full binary tree with n items has a height of logn.
Space Complexity
The amount of storage or working space a heap sort algorithm in Data Structures needs is called its space complexity. It is closely correlated with or proportionate to how much input the algorithm receives. Calculating how much space an algorithm's variables occupy will yield space complexity. The algorithm runs more quickly, and the less space it needs. It's also crucial to understand that space difficulty and temporal complexity are unrelated.
Implementation of Heap Sort
The following images show the implementation of Heap Sort in Java, Python, C and C++:
Heap sort in Java
public class HeapSort {
public void sort(int arr[]) {
int n = arr.length;
//Rearrange array (building heap)
for (int i = n / 2 - 1; i >= 0; i--) {
heapify(arr, n, i);
}
//Extract elements from heap one by one
for (int i = n - 1; i >= 0; i--) {
//Current root moved to the end
int tmp = arr[0];
arr[0] = arr[i];
arr[i] = tmp;
heapify(arr, i, 0);//calling max heapify on the heap reduced
}
}
void heapify(int arr[], int n, int i) {
int max = i; //Initialize max as root
int leftChild = 2 * i + 1;
int rightChild = 2 * i + 2;
//If left child is greater than root
if (leftChild < n && arr[leftChild] > arr[max])
max = leftChild;
//If right child is greater than max
if (rightChild < n && arr[rightChild] > arr[max])
max = rightChild;
//If max is not root
if (max != i) {
int swap = arr[i];
arr[i] = arr[max];
arr[max] = swap;
//heapify the affected sub-tree recursively
heapify(arr, n, max);
}
}
//print size of array n using utility function
static void display(int arr[]) {
int n = arr.length;
for (int i = 0; i < n; ++i)
System.out.print(arr[i] + " ");
System.out.println();
}
//Driver code
public static void main(String args[]) {
int arr[] = {11, 34, 9, 5, 16, 10};
HeapSort hs = new HeapSort();
System.out.println("Original array:");
display(arr);
hs.sort(arr);
System.out.println("Sorted array:");
display(arr);
}
}
Output of the program:
Original array:
11 34 9 5 16 10
Sorted array:
5 9 10 11 16 34
Heap sort in Python
def heapify(arr, n, i):
largest = i #Initialize max as root
l = 2 * i + 1
r = 2 * i + 2
//If left child is greater than root
if l < n and arr[i] < arr[l]:
largest = l
//If right child is greater than max
if r < n and arr[largest] < arr[r]:
largest = r
//If max is not root
if largest != i:
arr[i], arr[largest] = arr[largest], arr[i]
heapify(arr, n, largest) //heapify the root
//Main function to perform heap sort
def heapSort(arr):
n = len(arr)
#Build MaxHeap
for i in range(n //2, -1, -1):
heapify(arr, n, i)
#Extract elements from heap one by one
for i in range(n - 1, 0, -1):
arr[i], arr[0] = arr[0], arr[i]
heapify(arr, i, 0)
#Driver code
arr = [11, 34, 9, 5, 16, 10]
n = len(arr)
print("Original array:")
for i in range(n):
print("%d " % arr[i], end = '')
heapSort(arr)
print("Sorted array:")
for i in range(n):
print("%d " % arr[i], end = '')
Output of the program:
Original array:
11 34 9 5 16 10
Sorted array:
5 9 10 11 16 34
Heap sort in C
#include
void swap(int *a, int *b) {
int tmp = *a;
*a = *b;
*b = tmp;
}
void heapify(int arr[], int n, int i) {
int max = i; //Initialize max as root
int leftChild = 2 * i + 1;
int rightChild = 2 * i + 2;
//If left child is greater than root
if (leftChild < n && arr[leftChild] > arr[max])
max = leftChild;
//If right child is greater than max
if (rightChild < n && arr[rightChild] > arr[max])
max = rightChild;
//If max is not root
if (max != i) {
swap(&arr[i], &arr[max]);
//heapify the affected sub-tree recursively
heapify(arr, n, max);
}
}
//Main function to perform heap sort
void heapSort(int arr[], int n) {
//Rearrange array (building heap)
for (int i = n / 2 - 1; i >= 0; i--)
heapify(arr, n, i);
//Extract elements from heap one by one
for (int i = n - 1; i >= 0; i--) {
swap(&arr[0], &arr[i]); //Current root moved to the end
heapify(arr, i, 0); //calling max heapify on the heap reduced
}
}
//print size of array n using utility function
void display(int arr[], int n) {
for (int i = 0; i < n; ++i)
printf("%d ", arr[i]);
printf("\n");
}
//Driver code
int main() {
int arr[] = {11, 34, 9, 5, 16, 10};
int n = sizeof(arr) / sizeof(arr[0]);
printf("Original array:\n");
display(arr, n);
heapSort(arr, n);
printf("Sorted array:\n");
display(arr, n);
}
Output of the program:
Original array:
11 34 9 5 16 10
Sorted array:
5 9 10 11 16 34
Heap sort in C++
#include
using namespace std;
void heapify(int arr[], int n, int i) {
int max = i; //Initialize max as root
int leftChild = 2 * i + 1;
int rightChild = 2 * i + 2;
//If left child is greater than root
if (leftChild < n && arr[leftChild] > arr[max])
max = leftChild;
//If right child is greater than max
if (rightChild < n && arr[rightChild] > arr[max])
max = rightChild;
//If max is not root
if (max != i) {
swap(arr[i], arr[max]);
heapify(arr, n, max); //heapify the affected sub-tree recursively
}
}
//Main function to perform heap sort
void heapSort(int arr[], int n) {
//Rearrange array (building heap)
for (int i = n / 2 - 1; i >= 0; i--)
heapify(arr, n, i);
//Extract elements from heap one by one
for (int i = n - 1; i >= 0; i--) {
swap(arr[0], arr[i]); //Current root moved to the end
heapify(arr, i, 0); //calling max heapify on the heap reduced
}
}
//print size of array n using utility function
void display(int arr[], int n) {
for (int i = 0; i < n; ++i)
cout << arr[i] << " ";
cout << "\n";
}
//Driver code
int main() {
int arr[] = {11, 34, 9, 5, 16, 10};
int n = sizeof(arr) / sizeof(arr[0]);
cout << "Original array:\n";
display(arr, n);
heapSort(arr, n);
cout << "Sorted array:\n";
display(arr, n);
}
Output of the program:
Original array:
11 34 9 5 16 10
Sorted array:
5 9 10 11 16 34
Advantages of Heap Sort in Data Structure
Heap sort has the following advantages:
- Efficiency – While other heap sort algorithms in Data Structures may become exponentially slower as the number of items to sort rises, the amount of time needed to complete a heap sort increases logarithmically. This sorting method is quite effective.
- Memory Usage – Memory use is modest since it requires the minimum amount of memory needed to store the initial list of things to be sorted.
- Simplicity – Since it does not employ complex computer science ideas like recursion, it is easier to comprehend than other similarly effective sorting algorithms.
Disadvantages of Heap Sort in Data Structure
There are certain disadvantages of heap sort, such as:
- Costly: Heap sort is usually expensive.
- Unstable: The heap sort is erratic because its relative order could change.
- Efficient: Heap sort is not particularly effective when dealing with complicated data.
Applications of Heap Sort in Data Structure
The Heap Sort in Data Structure is used in the applications listed below.
Heaps are a component in priority queue creation. We can delete, insert, look for the element by prioritising it, or extract and insert it with priority with the priority queue by taking O (log N) time. While Heap Sort in Data Structure and heap like the AVL trees, Red-Black tree, and Binary Search Tree may perform similar functions, they are more sophisticated.
An actual instance of the application of priority queues- This kind of line might be employed when clients who want quick service are given preference over those who arrive early. For instance, clients with a modest balance may be preferred in a licensing centre.
The priority queues developed using heap data structure have more complex usage in graph algorithms, which can minimise the average waiting time for all clients. Dijkstra's algorithm, Prims, Huffman coding, and BFS algorithm are a few of them.
The Heap Sort in Data Structure makes it simple and quick to determine which member in an array is the kth smallest or biggest.
Systems that deal with security and embedded systems use heap sort.
The heap sort method makes advantage of the heap data structure and is simple and effective.A reliable sorting algorithm known as the Heap Sort in Data Structure has a worst-case time complexity of O (nlogn).
Conclusion
In conclusion, a Heap Sort in Data Structure is a useful tool for organising huge datasets and a crucial technique for anybody learning about Heap Sort in Data Structure and algorithms to comprehend. The Learner Success Team at Hero Vired will ensure you get the training needed to master data science. Learn from experts who will assist you as you progress through this certification programme. Want to know about stack in data structures? Read the blog!
Author's Note
Hero Vired is a leading learn tech firm that connects with top universities to deliver programmes relevant to the industry and develop tomorrow's change-makers.
