Data Structures & Algorithms
Table of content:
- What Is Data Structure?
- Key Features Of Data Structures
- Basic Terminologies Related To Data Structures
- Types Of Data Structures
- What Are Linear Data Structures & Its Types?
- What Are Non-Linear Data Structures & Its Types?
- Importance Of Data Structure In Programming
- Basic Operations On Data Structures
- Applications Of Data Structures
- Real-Life Applications Of Data Structures
- Linear Vs. Non-linear Data Structures
- What Are Algorithms? The Difference Between Data Structures & Algorithms
- Conclusion
- Frequently Asked Questions
Table of content:
- What Is Asymptotic Notation?
- How Asymptotic Notation Helps In Analyzing Performance
- Types Of Asymptotic Notation
- Big-O Notation (O)
- Omega Notation (Ω)
- Theta Notation (Θ)
- Little-O Notation (o)
- Little-Omega Notation (ω)
- Summary Of Asymptotic Notations
- Real-World Applications Of Asymptotic Notation
- Conclusion
- Frequently Asked Questions
Table of content:
- Understanding Big O Notation
- Types Of Time Complexity
- Space Complexity In Big O Notation
- How To Determine Big O Complexity
- Best, Worst, And Average Case Complexity
- Applications Of Big O Notation
- Conclusion
- Frequently Asked Questions
Table of content:
- What Is Time Complexity?
- Why Do You Need To Calculate Time Complexity?
- The Time Complexity Algorithm Cases
- Time Complexity: Different Types Of Asymptotic Notations
- How To Calculate Time Complexity?
- Time Complexity Of Sorting Algorithms
- Time Complexity Of Searching Algorithms
- Writing And optimizing An algorithm
- What Is Space Complexity And Its Significance?
- Relation Between Time And Space Complexity
- Conclusion
Table of content:
- What Is Linear Data Structure?
- Key Characteristics Of Linear Data Structures
- What Are The Types Of Linear Data Structures?
- What Is An Array Linear Data Structure?
- What Are Linked Lists Linear Data Structure?
- What Is A Stack Linear Data Structure?
- What Is A Queue Linear Data Structure?
- Most Common Operations Performed in Linear Data Structures
- Advantages Of Linear Data Structures
- What Is Nonlinear Data Structure?
- Difference Between Linear & Non-Linear Data Structures
- Who Uses Linear Data Structures?
- Conclusion
- Frequently Asked Questions
Table of content:
- What is a linear data structure?
- What is a non-linear data structure?
- Difference between linear data structure and non-linear data structure
- FAQs about linear and non-linear data structures
Table of content:
- What Is Sorting In Data Structures?
- What Is Bubble Sort?
- What Is Selection Sort?
- What Is Insertion Sort?
- What Is Merge Sort?
- What Is Quick Sort?
- What Is Heap Sort?
- What Is Radix Sort?
- What Is Bucket Sort?
- What Is Counting Sort?
- What Is Shell Sort?
- Choosing The Right Sorting Algorithm
- Applications Of Sorting
- Conclusion
- Frequently Asked Questions
Table of content:
- Understanding Bubble Sort Algorithm
- Bubble Sort Algorithm
- Implementation Of Bubble Sort In C++
- Time And Space Complexity Analysis Of Bubble Sort Algorithm
- Advantages Of Bubble Sort Algorithm
- Disadvantages Of Bubble Sort Algorithm
- Applications of Bubble Sort Algorithms
- Conclusion
- Frequently Asked Questions
Table of content:
- Understanding The Merge Sort Algorithm
- Algorithm For Merge Sort
- Implementation Of Merge Sort In C++
- Time And Space Complexity Analysis Of Merge Sort
- Advantages And Disadvantages Of Merge Sort
- Applications Of Merge Sort
- Conclusion
- Frequently Asked Questions
Table of content:
- Understanding The Selection Sort Algorithm
- Algorithmic Approach To Selection Sort
- Implementation Of Selection Sort Algorithm
- Complexity Analysis Of Selection Sort
- Comparison Of Selection Sort With Other Sorting Algorithms
- Advantages And Disadvantages Of Selection Sort
- Application Of Selection Sort
- Conclusion
- Frequently Asked Questions
Table of content:
- What Is Insertion Sort Algorithm?
- How Does Insertion Sort Work? (Step-by-Step Explanation)
- Implementation of Insertion Sort in C++
- Time And Space Complexity Of Insertion Sort
- Applications Of Insertion Sort Algorithm
- Comparison With Other Sorting Algorithms
- Conclusion
- Frequently Asked Questions
Table of content:
- Understanding Quick Sort Algorithm
- Step-By-Step Working Of Quick Sort
- Implementation Of Quick Sort Algorithm
- Time Complexity Analysis Of Quick Sort
- Advantages Of Quick Sort Algorithm
- Disadvantages Of Quick Sort Algorithm
- Applications Of Quick Sort Algorithm
- Conclusion
- Frequently Asked Questions
Table of content:
- Understanding The Heap Data Structure
- Working Of Heap Sort Algorithm
- Implementation Of Heap Sort Algorithm
- Advantages Of Heap Sort
- Disadvantages Of Heap Sort
- Real-World Applications Of Heap Sort
- Conclusion
- Frequently Asked Questions
Table of content:
- Understanding The Counting Sort Algorithm
- Conditions For Using Counting Sort Algorithm
- How Counting Sort Algorithm Works?
- Implementation Of Counting Sort Algorithm
- Time And Space Complexity Analysis Of Counting Sort
- Comparison Of Counting Sort With Other Sorting Algorithms
- Advantages Of Counting Sort Algorithm
- Disadvantages Of Counting Sort Algorithm
- Applications Of Counting Sort Algorithm
- Conclusion
- Frequently Asked Questions
Table of content:
- Understanding Shell Sort Algorithm
- Working Of Shell Sort Algorithm
- Implementation Of Shell Sort Algorithm
- Time Complexity Analysis Of Shell Sort Algorithm
- Advantages Of Shell Sort Algorithm
- Disadvantages Of Shell Sort Algorithm
- Applications Of Shell Sort Algorithm
- Conclusion
- Frequently Asked Questions
Table of content:
- What Is The Binary Search Algorithm?
- Conditions For Using Binary Search
- Steps For Implementing Binary Search
- Iterative Method For Binary Search With Implementation Examples
- Recursive Method For Binary Search
- Complexity Analysis Of Binary Search Algorithm
- Iterative Vs. Recursive Implementation Of Binary Search
- Advantages & Disadvantages Of Binary Search
- Practical Applications & Real-World Examples Of Binary Search
- Conclusion
- Frequently Asked Questions
Table of content:
- What Is Linked List In Data Structures?
- Types Of Linked Lists In Data Structures
- Linked List Operations In Data Structures
- Advantages And Disadvantages Of Linked Lists In Data Structures
- Comparison Of Linked Lists And Arrays
- Applications Of Linked Lists
- Conclusion
- Frequently Asked Questions
Table of content:
- What Is A Singly Linked List In Data Structure?
- Insertion Operations On Singly Linked Lists
- Deletion Operation On Singly Linked List
- Searching For Elements In Single Linked List
- Calculating Length Of Single Linked List
- Practical Applications Of Singly Linked Lists In Data Structure
- Common Problems With Singly Linked Lists
- Conclusion
- Frequently Asked Questions (FAQ)
Table of content:
- What Is A Linked List?
- Reverse A Linked List
- How To Reverse A Linked List? (Approaches)
- Recursive Approach To Reverse A Linked List
- Iterative Approach To Reverse A Linked List
- Using Stack To Reverse A Linked List
- Complexity Analysis Of Different Approaches To Reverse A Linked List
- Conclusion
- Frequently Asked Questions
Table of content:
- What Is A Graph Data Structure?
- Importance Of Graph Data Structures
- Types Of Graphs In Data Structure
- Types Of Graph Algorithms
- Application Of Graphs In Data Structures
- Challenges And Complexities In Graphs
- Conclusion
- Frequently Asked Questions
Table of content:
- What Is Tree Data Structure?
- Terminologies Of Tree Data Structure:
- Different Types Of Tree Data Structures
- Basic Operations On Tree Data Structure
- Applications Of Tree Data Structures
- Comparison Of Trees, Graphs, And Linear Data Structures
- Advantages Of Tree Data Structure
- Disadvantages Of Tree Data Structure
- Conclusion
- Frequently Asked Questions
Table of content:
- What Is Dynamic Programming?
- Real-Life Example: The Jigsaw Puzzle Analogy
- How To Solve A Problem Using Dynamic Programming?
- Dynamic Programming Algorithm Techniques
- Advantages Of Dynamic Programming
- Disadvantages Of Dynamic Programming
- Applications Of Dynamic Programming
- Conclusion
- Frequently Asked Questions
Table of content:
- Understanding The Sliding Window Algorithm
- How Does The Sliding Window Algorithm Works?
- How To Identify Sliding Window Problems?
- Fixed-Size Sliding Window Example: Maximum Sum Subarray Of Size k
- Variable-Size Sliding Window Example: Smallest Subarray With A Given Sum
- Advantages Of Sliding Window Technique
- Disadvantages Of Sliding Window Technique
- Conclusion
- Frequently Asked Questions
Table of content:
- Introduction To Data Structures
- Data Structures Interview Questions: Basics
- Data Structures Interview Questions: Intermediate
- Data Structures Interview Questions: Advanced
- Conclusion
Merge Sort Algorithm | Working, Applications & More (+Examples)

Sorting is a fundamental operation in computer science, essential for organizing data efficiently. Among various sorting algorithms, Merge Sort stands out due to its divide-and-conquer approach, ensuring stable and efficient sorting. Developed by John von Neumann in 1945, Merge Sort recursively divides an array into smaller subarrays, sorts them individually, and then merges them back together in a sorted order.
In this article, we will explore the working of Merge Sort, its implementation, time complexity, and why it is preferred over other sorting techniques in certain scenarios.
Understanding The Merge Sort Algorithm
Merge Sort in data structures is based on the Divide and Conquer approach, which breaks a complex problem into smaller subproblems, solves them individually, and then combines the results.
Real-Life Analogy: Organizing a Messy Bookshelf
Imagine you have a huge pile of books scattered randomly, and you want to arrange them in alphabetical order. Instead of sorting the entire pile at once, you can use a systematic approach:
- Divide: Split the pile into two smaller halves.
- Conquer: Keep dividing each half further until each section contains just one book (which is already sorted by itself).
- Combine: Start merging two small piles at a time, ensuring they stay in order, until all books are sorted in a single, neatly arranged bookshelf.
This is exactly how Merge Sort works with an array of numbers!
Explanation Of The Divide-and-Conquer Approach In Merge Sort
- Divide: The array is recursively divided into two halves until each subarray contains only one element (which is naturally sorted).
- Conquer: Each pair of small sorted arrays is merged together in sorted order.
- Combine: The merging continues until we get a fully sorted array.
For example, let’s sort the array [8, 3, 5, 1, 7, 2, 6, 4] using Merge Sort:
- Divide:
[8, 3, 5, 1] [7, 2, 6, 4]
[8, 3] [5, 1] [7, 2] [6, 4]
[8] [3] [5] [1] [7] [2] [6] [4]
(Each number is now a single-element array.)
- Conquer (Merge Step):
[3, 8] [1, 5] [2, 7] [4, 6]
[1, 3, 5, 8] [2, 4, 6, 7]
[1, 2, 3, 4, 5, 6, 7, 8]
Each merge step ensures that the resulting array remains sorted.
Why is Divide and Conquer Efficient?
- Instead of comparing every element with every other element (O(n²) time in Bubble Sort or Insertion Sort), Merge Sort cleverly splits and merges efficiently in O(n log n) time.
- The systematic merging ensures stability, meaning equal elements retain their relative order, which is crucial in certain applications like database sorting.
Thus, Merge Sort leverages Divide and Conquer to sort data efficiently, just like organizing a messy bookshelf in small, manageable steps!
Algorithm For Merge Sort
The main steps in the merge sort algorithm are:
1. Splitting the Array (Divide Phase)
- We keep recursively dividing the array into two halves until each subarray contains a single element (which is inherently sorted).
2. Sorting and Merging the Subarrays (Conquer & Combine Phase)
- Once we reach the smallest subarrays, we start merging them back in a sorted order.
- This merging process compares elements from two sorted subarrays and arranges them in the correct order.
Pseudocode Representation:
MERGE_SORT(arr, left, right)
1. If left < right:
2. Find the middle index: mid = (left + right) / 2
3. Recursively sort the first half: MERGE_SORT(arr, left, mid)
4. Recursively sort the second half: MERGE_SORT(arr, mid + 1, right)
5. Merge the sorted halves: MERGE(arr, left, mid, right)MERGE(arr, left, mid, right)
1. Create two temporary arrays:
Left subarray = arr[left...mid]
Right subarray = arr[mid+1...right]
2. Maintain three pointers for Left, Right, and Merged arrays.
3. Compare elements from Left and Right subarrays:
- Copy the smaller element into the merged array.
- Move the corresponding pointer forward.
4. If elements remain in either subarray, copy them into the merged array.
Implementation Of Merge Sort In C++
Here’s a C++ program to implement Merge Sort with a step-by-step explanation:
Code Example:
#include <iostream>
using namespace std;
// Function to merge two sorted subarrays
void merge(int arr[], int left, int mid, int right) {
int n1 = mid - left + 1; // Size of left subarray
int n2 = right - mid; // Size of right subarray
// Create temporary arrays
int leftArr[n1], rightArr[n2];
// Copy data into temporary arrays
for (int i = 0; i < n1; i++)
leftArr[i] = arr[left + i];
for (int j = 0; j < n2; j++)
rightArr[j] = arr[mid + 1 + j];
// Merge the temporary arrays back into arr[left...right]
int i = 0, j = 0, k = left;
while (i < n1 && j < n2) {
if (leftArr[i] <= rightArr[j]) {
arr[k] = leftArr[i];
i++;
} else {
arr[k] = rightArr[j];
j++;
}
k++;
}
// Copy remaining elements of leftArr[], if any
while (i < n1) {
arr[k] = leftArr[i];
i++;
k++;
}
// Copy remaining elements of rightArr[], if any
while (j < n2) {
arr[k] = rightArr[j];
j++;
k++;
}
}
// Merge Sort function (recursively divides and sorts)
void mergeSort(int arr[], int left, int right) {
if (left < right) {
int mid = left + (right - left) / 2;
// Recursively sort first and second halves
mergeSort(arr, left, mid);
mergeSort(arr, mid + 1, right);
// Merge the sorted halves
merge(arr, left, mid, right);
}
}
// Function to print an array
void printArray(int arr[], int size) {
for (int i = 0; i < size; i++)
cout << arr[i] << " ";
cout << endl;
}
// Main function to test Merge Sort
int main() {
int arr[] = {8, 3, 5, 1, 7, 2, 6, 4};
int size = sizeof(arr) / sizeof(arr[0]);
cout << "Original array: ";
printArray(arr, size);
mergeSort(arr, 0, size - 1);
cout << "Sorted array: ";
printArray(arr, size);
return 0;
}
Output:
Original array: 8 3 5 1 7 2 6 4
Sorted array: 1 2 3 4 5 6 7 8
Explanation:
In the above code example-
- We begin by including the <iostream> header file, which allows us to use standard input and output operations.
- Then we use namespace std; to avoid prefixing std:: before standard library elements.
- The merge() function is responsible for merging two sorted subarrays into a single sorted subarray.
- We calculate the sizes of the two subarrays using mid - left + 1 for the left subarray and right - mid for the right subarray.
- Temporary arrays leftArr and rightArr store elements of the left and right subarrays, respectively.
- We copy elements from the original array into these temporary arrays.
- A while loop merges the two temporary arrays back into the original array while maintaining the sorted order.
- If elements remain in either subarray, we copy them back into the original array.
- The mergeSort function recursively divides the array into halves until we reach single-element subarrays.
- Once divided, the function merges the sorted subarrays back using the merge function.
- The printArray function prints the elements of an array, separating them with spaces.
- In main() function, we define an integer array {8, 3, 5, 1, 7, 2, 6, 4} and calculate its size.
- We print the original array using printArray.
- We call mergeSort, passing the array and its boundaries (0 to size - 1).
- After sorting, we print the sorted array using printArray.
- The program follows the divide-and-conquer approach, making recursive calls and merging sorted subarrays efficiently.
Time And Space Complexity Analysis Of Merge Sort
Merge Sort follows the Divide and Conquer approach, recursively breaking down the array and merging sorted subarrays. Below is a detailed analysis of its time and space complexity:
Case |
Time Complexity |
Explanation |
Best Case |
O(n log n) |
Occurs when the array is already sorted. The algorithm still divides and merges, leading to O(n log n). |
Average Case |
O(n log n) |
The array is randomly ordered. The division process takes O(log n), and merging takes O(n), resulting in O(n log n). |
Worst Case |
O(n log n) |
Even in the worst case (reverse-sorted array), the divide and merge steps remain O(n log n). |
Space Complexity |
O(n) |
Extra space is required to store temporary subarrays during merging. |
Key Takeaways
- Merge Sort maintains consistent O(n log n) performance across all cases.
- It is not in-place due to its O(n) space requirement, making it less memory-efficient than Quick Sort.
- Suitable for sorting large datasets and linked lists due to its stability and efficiency.
Advantages And Disadvantages Of Merge Sort
Merge Sort is a widely used sorting algorithm known for its efficiency and stability. However, it also has some drawbacks.
Advantages Of Merge Sort
Advantage |
Explanation |
Consistent O(n log n) Time Complexity |
Unlike Quick Sort, which has a worst case of O(n²), Merge Sort always runs in O(n log n), making it predictable. |
Stable Sorting Algorithm |
Maintains the relative order of equal elements, which is useful in applications like database sorting. |
Efficient for Large Datasets |
Works well with large arrays and linked lists since it divides the problem into smaller parts and merges them efficiently. |
Works Well with External Sorting |
Since it processes data in chunks, it is useful for sorting very large files stored in external memory (e.g., disk-based sorting). |
Disadvantages Of Merge Sort Algorithm
Disadvantage |
Explanation |
Higher Space Complexity (O(n)) |
Requires extra O(n) space for temporary subarrays, making it not memory-efficient for large in-memory sorting. |
Slower for Small Arrays |
Other algorithms like Insertion Sort are faster for small datasets due to lower overhead. |
Not an In-Place Algorithm |
Unlike Quick Sort, which sorts within the original array, Merge Sort requires additional space, making it less suitable for memory-constrained environments. |
Applications Of Merge Sort
Merge Sort is widely used in real-world applications due to its efficiency and stability. Here are some key areas where it is applied:
- Sorting Large Datasets – Efficiently sorts massive datasets, especially when dealing with millions of elements.
- External Sorting (Disk-Based Sorting) – Used when the data does not fit into memory, such as sorting large log files or databases stored on hard drives or SSDs.
- Linked List Sorting – Performs better than Quick Sort for linked lists since it doesn’t require random access and works efficiently in O(n log n) time.
- Stable Sorting in Databases – Ensures that records with the same key retain their original order, making it ideal for database management systems (DBMS).
- Multi-Threaded Sorting – Works well in parallel computing environments since it can divide the problem into smaller parts and sort them concurrently.
- Genomic and Bioinformatics Applications – Used in DNA sequencing and other computational biology tasks where sorting huge amounts of genetic data is required.
- Sorting in Graphics and Computer Vision – Helps in image processing tasks, such as sorting pixels by intensity levels for efficient rendering and filtering.
Conclusion
Merge Sort is a powerful and efficient sorting algorithm that follows the Divide and Conquer approach. With its consistent O(n log n) time complexity and stability, it is widely used in applications such as large dataset sorting, linked lists, external sorting, and database management. However, its O(n) space complexity makes it less memory-efficient compared to in-place sorting algorithms like Quick Sort.
Think of Merge Sort like assembling a puzzle—we break it into smaller pieces, sort them individually, and then carefully put them back together to form the final sorted result. While it may not always be the fastest choice for small datasets, its reliability and scalability make it an excellent choice for handling large and complex sorting problems.
Frequently Asked Questions
Q. Why is Merge Sort considered a stable sorting algorithm?
Merge Sort is stable because it maintains the relative order of equal elements in the original array. When merging two sorted subarrays, if two elements have the same value, the element from the left subarray is placed first, preserving their initial order. This property is important in applications like database sorting, where maintaining record order is necessary.
Q. How does Merge Sort use the Divide and Conquer approach?
Merge Sort follows the Divide and Conquer strategy in three steps:
- Divide – The array is recursively split into two halves until each subarray has only one element.
- Conquer – Each subarray is sorted independently.
- Combine (Merge) – The sorted subarrays are merged back in order to produce the final sorted array.
This approach ensures efficient sorting and allows for parallel execution in multi-threaded environments.
Q. Why does Merge Sort always have O(n log n) time complexity?
Merge Sort maintains O(n log n) complexity in all cases because:
- The Divide step always takes log n time, as the array is halved in each recursive call.
- The Merge step takes O(n) time to combine two sorted subarrays.
- Since both steps run for every level of recursion, the overall complexity is O(n log n).
Unlike Quick Sort, which degrades to O(n²) in the worst case, Merge Sort remains consistently efficient.
Q. Why is Merge Sort not an in-place sorting algorithm?
An in-place algorithm sorts data without requiring significant extra memory. Merge Sort algorithm, however, needs O(n) extra space to store temporary subarrays during merging. Because it creates copies of subarrays, it does not modify the original array in place, making it less memory-efficient compared to algorithms like Quick Sort or Heap Sort.
Q. In which scenarios is Merge Sort preferred over Quick Sort?
Merge Sort Algorithm is preferred over Quick Sort Algorithm in the following cases:
- Sorting Linked Lists – Since Merge Sort doesn’t require random access to elements, it performs better on linked lists than Quick Sort.
- Stable Sorting Required – When maintaining the relative order of equal elements is essential, Merge Sort is a better choice.
- Sorting Large Files (External Sorting) – Merge Sort works well for disk-based sorting, where data is too large to fit into memory.
- Worst-Case Performance Matters – Merge Sort guarantees O(n log n) in all cases, whereas Quick Sort can degrade to O(n²) in the worst case.
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I’m a Computer Science graduate with a knack for creative ventures. Through content at Unstop, I am trying to simplify complex tech concepts and make them fun. When I’m not decoding tech jargon, you’ll find me indulging in great food and then burning it out at the gym.
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