What is a Merge Sort Algorithm? How Does it Work?

Merge Sort Algorithm Introduction 

The category Sorting algorithms based on comparison strategies includes the well-liked and effective sorting algorithm known as Merge Sort. John von Neyy made the initial presentation of it in 1945. On both small and large datasets, the method performs consistently and steadily. Merge sort employs a divide and conquer method by separating the input array into smaller sub-arrays, creating the final sorted result by recursively sorting the items, then merging the items one more. 

How Merge Sort Works?

Merge Sort is a popular sorting algorithm that follows the Divide and Conquer approach to sort an array or a list of elements. It works as follows:

Divide and Conquer Approach:

The Merge formula Sort’s ability to sort enormous datasets effectively results from its divide and conquer method. Three easy steps can be used, to sum up the procedure:

Step 1 – Divide: Up until each sub-array has just one element, the unsorted array is split into two equal portions. Up until there are no more divisions, this procedure is continued in a recursive fashion.

Step 2 – Conquer: The particular-element sub-arrays are assumed to be sorted by default because a single element is always sorted.

Step 3 – Merge: Recombining the sorted sub-arrays places the elements of the larger sorting array in the correct order. During the combining process, the components of the two sub-arrays are compared and then arranged chronologically. When comparing the elements in the two arrays, the algorithm chooses the smaller element, inserting it into the new, sorted array. This process is repeated until all of the components of both sub-arrays are combined into the ultimate sorting of an array.

Merge Step:

An essential component of the merging Sort algorithm is the merging stage. The method successfully merges two sorted sub-arrays into one sorted array at this stage. It entails comparing and ordering the components of both sub-arrays in chronological order.

The algorithm compares the components at these two points, one for each sub-array. The matching pointer is advanced while the smaller components are transferred to the new sorted array. This process is repeated until all of the components of both sub-arrays are combined into the ultimate sorting of an array.

Merge Sort Time Complexity:

To comprehend merge sort time complexity and performance on various datasets, it is essential to grasp its time complexity. Big O notation is used to express the temporal complexity of Merge Sort.

• Best Case: The maximum time complexity of Merge Sort, where ‘n’ is the number of elements in the input array, is O(n log n). When the input array is already resolved or almost sorted, this happens.
• Worst Case: Merge Sort still has an O(n log n) time complexity, even in the worst-case situation. This is due to the algorithm’s constant splitting of the input array in half and recursive sorting of the two halves. As a result, the best-case time complexity is also the worst-case time complexity.
• Average Case: Merge Sort has an average-case time complexity of O(n log n). Merge Sort is frequently chosen for big datasets because of its superior performance over quadratic sorting algorithms.

Implementing Merge Sort Python

The Merge Sort algorithm is implemented in Python in the following manner:

def merge_sort(arr):

if len(arr) <= 1:

     return arr

mid = len(arr) // 2

left_half = arr[:mid]

right_half = arr[mid:]

left_half = merge_sort(left_half)

right_half = merge_sort(right_half)

return merge(left_half, right_half)

def merge(left, right):

result = []

left_idx, right_idx = 0, 0

while left_idx < len(left) and right_idx < len(right):

     if left[left_idx] < right[right_idx]:

            result.append(left[left_idx])

         left_idx += 1

     else:

            result.append(right[right_idx])

         right_idx += 1

result += left[left_idx:]

result += right[right_idx:]

return result


# Example usage:

arr = [38, 27, 43, 3, 9, 82, 10]

sorted_arr = merge_sort(arr)

print(sorted_arr)

```

C Merge Sort Program

Here is a detailed description of how the Merge Sort algorithm is implemented in C:

#include <stdio.h>

void merge(int arr[], int left, int mid, int right) {

int i, j, k;

int n1 = mid - left + 1;

int n2 = right - mid;

int L[n1], R[n2];

for (i = 0; i < n1; i++)

     L[i] = arr[left + i];

for (j = 0; j < n2; j++)

     R[j] = rr[mid + 1 + j];

i = 0;

j = 0;

k = left;

while (i < n1 && j < n2) {

     if (L[i] <= R[j]) {

         arr[k] = L[i];

         i++;

     }

     else {

         arr[k] = R[j];

         j++;

     }

     k++;

}

while (i < n1) {

     arr[k] = L[i];

     i++;

     k++;

}

while (j < n2) {

     arr[k] = R[j];

     j++;

     k++;

}

} 

  void merge_sort(int arr[], int left, int right) {

if (left < right) {

     int mid = left + (right - left) / 2;

 

     merge_sort(arr, left, mid);

     merge_sort(arr, mid + 1, right);

 

     merge(arr, left, mid, right);

}

}

 

int main() {

int arr[] = {38, 27, 43, 3, 9, 82, 10};

int n = sizeof(arr) / sizeof(arr[0]);

 

merge_sort(arr, 0, n - 1);

 

printf("Sorted array: ");

for (int i = 0; i < n; i++)

     printf("%d ", arr[i]);

 

return 0;

}

Merge Sort in Data Structures

Due to its effectiveness and reliability, merge sort is extensively employed in different data structures. It is frequently used to sort linked lists, which presents difficulties for more effective sorting algorithms like Quick Sort. Due to its divide and conquer approach, merge sort is a common data structure technique when dealing with linked lists.

An effective sorting algorithm that employs the divide-and-conquer strategy is merge sort. The unsorted list is split into single-element sublists before being merged back together during sorting. To create the final sorted list, the merging phase effectively joins sublists that have already been sorted. It is the best option for huge datasets because of its O(n log n) time complexity. However, it needs more RAM to accommodate transient sublists while merging. Merge Sort is well-liked overall for its reliability, consistency, and ease of use.

Merge Sort Pseudocode

The following is a representation of the merge sort pseudocode:

merge_sort(arr):

if length of arr <= 1:

     return arr

 

mid = length of arr // 2

left_half = arr[:mid]

right_half = arr[mid:]

 

left_half = merge_sort(left_half)

right_half = merge_sort(right_half)

 

return merge(left_half, right_half)

 

merge(left, right):

result = []

left_idx, right_idx = 0, 0

 

while left_idx < length of left and right_idx < length of right:

     if left[left_idx] < right[right_idx]:

         append left[left_idx] to result

         left_idx += 1

     else:

         append right[right_idx] to result

         right_idx += 1

 

append remaining elements of left to result

append remaining elements of right to result

return result

Merge Sort Complexity

Time Complexity: The Merge Sort method is recursive, with time complexity given by the following recurrence relation: 

O(N log(N))

T(n) = 2T(n/2) θ(n)

The aforementioned recurrence can be resolved using either the Recurrence Tree approach or the Master method. Nlog(N) is the solution to the recurrence and fits into Case II of the Master Method. Merge sort always splits the array in half in all three scenarios (worst, average, and best), and because it requires linear time to join the two halves, its time complexity is Nlog(N).

Auxiliary Space: O(N), All elements in a merge sort are copied into a support array. N auxiliary spaces are therefore necessary for merge sort.

Comparing Merge Sort with Other Sorting Algorithms:

One of the most effective sorting algorithms, merge sort, is frequently contrasted with other well-known sorting algorithms like quick sort and heap sort in terms of time complexity.

Comparing Quick Sorting and Merge Sorting:

The efficient sorting algorithms Quick Sorting and Merge Sorting have an average time complexity of O(n log n). Merge Sort always maintains a worst-case time complexity of O(n log n), but If the pivot selection is poor, Quick Sort may have a worst-case time complexity of O(n2). Since worst-case performance is important, Merge Sort is more predictable and appropriate for real-world applications.

Compare Merge Sort and Heap Sort:

The average time complexity of Merge Sort and Heap Sort is O(n log n). Heap Sort is an in-place sorting algorithm, making it more memory-efficient than Merge Sort, which requires additional memory space for combining sub-arrays. Heap Sort’s speed on huge datasets may be impacted by the fact that it experiences more cache misses and unpredictable memory access patterns.

Merge Sort Uses

• Sorting large datasets: Due to its guaranteed worst-case time complexity of O(n log n), merge sort is especially well suited for sorting large datasets.
• External sorting: External sorting is often used when the data is too large to blend in memory.
• Custom sorting: Merge sort can be modified or modified to handle a wide range of input distributions, including partially, almost, and totally sorted data.

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Advantages of Merge Sort

• Stability: The relative order of equal elements in the input array is maintained using the stable sorting method known as merge sort.
• Guaranteed worst-case performance: Merge sort works well even on big datasets thanks to its worst-case time complexity of O(N logN).
• Parallelizable: Merge sort is a method that naturally scales to several processors or threads, making it easy to parallelize.

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Drawbacks of Merge Sort:

• Space complexity: During the sorting process, the combined sub-arrays from the merge sort must be stored in additional memory. 
• Not in place: Merge sort takes additional RAM to hold the sorted data because it is not an in-place sorting method. This might be a problem for programs when memory utilization is a problem.
• Not always optimal for small datasets: Merge sort has a higher time complexity than other sorting algorithms, such as insertion sort, for small datasets. This may cause performance to be slower for very tiny datasets.

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Applications of Merge Sort:

In many applications, sorting is a common operation. Merge Sort is the best option for maintaining the relative order of equal components due to its stability and predictable performance. Examples of typical applications include:

• Database administration: Sorting is essential for effective querying and indexing in database management systems.
• External Sorting: Merge Sort is a great choice for external sorting when data is kept on disk rather than in RAM because it can handle enormous datasets with little memory.
• Parallel Processing: Merge Sort is naturally parallelizable due to its divide-and-conquer structure, which enables effective sorting on multi-core processors and distributed systems.
• Merge Join: Merge Sort performs efficient merge joins when combining sorted data from two tables in database query optimization.

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Conclusion

In conclusion, the Divide and Conquer strategy is used in the Merge Sort algorithm, a strong and dependable sorting technique. Large datasets are efficiently sorted with a worst-case guaranteed time complexity of O(n log n). Because of its stability, the input array’s equal items are maintained in their relative order. Merge Sort is frequently used in many contexts, such as database management, external sorting, and parallel processing. Moreover, it is a vital tool for sorting algorithms.

It finds widespread use in various applications, including the Full Stack Software Development Bootcamp from upGrad, where efficiency, predictability, and stability are essential for students to master full-stack software development skills. Despite its space complexity and potential for inferior performance on extremely small datasets, Merge Sort is a popular option for sorting tasks, providing a reliable foundation for students at the boot camp to acquire the necessary expertise.