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C Program for Merge Sort

Like QuickSort, Merge Sort is a Divide and Conquer algorithm. It divides the input array into two halves, calls itself for the two halves, and then it merges the two sorted halves. The merge() function is used for merging two halves. The merge(arr, l, m, r) is a key process that assumes that arr[l..m] and arr[m+1..r] are sorted and merges the two sorted sub-arrays into one. 

Pseudocode :



• Declare left variable to 0 and right variable to n-1 
• Find mid by medium formula. mid = (left+right)/2
• Call merge sort on (left,mid)
• Call merge sort on (mid+1,rear)
• Continue till left is less than right
• Then call merge function to perform merge sort.

Algorithm:

Step 1: Start
Step 2: Declare an array and left, right, mid variable
Step 3: Perform merge function.
mergesort(array,left,right)
mergesort (array, left, right)
if left > right
return
mid= (left+right)/2
mergesort(array, left, mid)
mergesort(array, mid+1, right)
merge(array, left, mid, right)
Step 4: Stop

See the following C implementation for details.



MergeSort(arr[], l,  r)

If r > l

  • Find the middle point to divide the array into two halves: 
    • middle m = l + (r – l)/2
  • Call mergeSort for first half:   
    • Call mergeSort(arr, l, m)
  • Call mergeSort for second half:
    • Call mergeSort(arr, m + 1, r)
  • Merge the two halves sorted in step 2 and 3:
    • Call merge(arr, l, m, r)

How Merge sort Works?

To know the functioning of merge sort, lets consider an array arr[] = {38, 27, 43, 3, 9, 82, 10}

  • At first, check if the left index of array is less than the right index, if yes then calculate its mid point

 

  • Now, as we already know that merge sort first divides the whole array iteratively into equal halves, unless the atomic values are achieved. 
  • Here, we see that an array of 7 items is divided into two arrays of size 4 and 3 respectively.

 

  • Now, again find that is left index is less than the right index for both arrays, if found yes, then again calculate mid points for both the arrays.

 

  • Now, further divide these two arrays into further halves, until the atomic units of the array is reached and further division is not possible.

 

  • After dividing the array into smallest units, start merging the elements again based on comparison of size of elements
  • Firstly, compare the element for each list and then combine them into another list in a sorted manner.

 

  • After the final merging, the list looks like this:

 

Refer to below illustrations for further clarity:

The following diagram shows the complete merge sort process for an example array {38, 27, 43, 3, 9, 82, 10}. 

If we take a closer look at the diagram, we can see that the array is recursively divided into two halves till the size becomes

Merge Sort Program in C




// C program for Merge Sort
#include <stdio.h>
#include <stdlib.h>
  
// Merges two subarrays of arr[].
// First subarray is arr[l..m]
// Second subarray is arr[m+1..r]
void merge(int arr[], int l, int m, int r)
{
    int i, j, k;
    int n1 = m - l + 1;
    int n2 = r - m;
  
    // Create temp arrays
    int L[n1], R[n2];
  
    // Copy data to temp arrays
    // L[] and R[]
    for (i = 0; i < n1; i++)
        L[i] = arr[l + i];
    for (j = 0; j < n2; j++)
        R[j] = arr[m + 1 + j];
  
    // Merge the temp arrays back
    // into arr[l..r]
    // Initial index of first subarray
    i = 0;
  
    // Initial index of second subarray
    j = 0;
  
    // Initial index of merged subarray
    k = l;
    while (i < n1 && j < n2) {
        if (L[i] <= R[j]) {
            arr[k] = L[i];
            i++;
        }
        else {
            arr[k] = R[j];
            j++;
        }
        k++;
    }
  
    // Copy the remaining elements
    // of L[], if there are any
    while (i < n1) {
        arr[k] = L[i];
        i++;
        k++;
    }
  
    // Copy the remaining elements of
    // R[], if there are any
    while (j < n2) {
        arr[k] = R[j];
        j++;
        k++;
    }
}
  
// l is for left index and r is
// right index of the sub-array
// of arr to be sorted
void mergeSort(int arr[], int l, int r)
{
    if (l < r) {
        // Same as (l+r)/2, but avoids
        // overflow for large l and r
        int m = l + (r - l) / 2;
  
        // Sort first and second halves
        mergeSort(arr, l, m);
        mergeSort(arr, m + 1, r);
  
        merge(arr, l, m, r);
    }
}
  
// UTILITY FUNCTIONS
// Function to print an array
void printArray(int A[], int size)
{
    int i;
    for (i = 0; i < size; i++)
        printf("%d ", A[i]);
    printf("\n");
}
  
// Driver code
int main()
{
    int arr[] = { 12, 11, 13, 5, 6, 7 };
    int arr_size = sizeof(arr) / sizeof(arr[0]);
  
    printf("Given array is \n");
    printArray(arr, arr_size);
  
    mergeSort(arr, 0, arr_size - 1);
  
    printf("\nSorted array is \n");
    printArray(arr, arr_size);
    return 0;
}

Output
Given array is 
12 11 13 5 6 7 

Sorted array is 
5 6 7 11 12 13 

Time Complexity: O(nlog(n))

Sorting arrays on different machines. Merge Sort is a recursive algorithm and time complexity can be expressed as following recurrence relation. 

T(n) = 2T(n/2) + θ(n)

The above recurrence can be solved either using the Recurrence Tree method or the Master method. It falls in case II of Master Method and the solution of the recurrence is θ(nLogn). Time complexity of Merge Sort is  θ(nLogn) in all 3 cases (worst, average and best) as merge sort always divides the array into two halves and takes linear time to merge two halves.

Auxiliary Space: O(n)

In merge sort, all elements are copied into an auxiliary array so N auxiliary space is required for merge sort.

Algorithmic Paradigm: Divide and Conquer

Is Merge Sort In-Place?

 No in a typical implementation

Is Merge sort Stable?

Yes, merge sort is stable. 

Applications of Merge Sort: 

Drawbacks of Merge Sort:

Please refer complete article on Merge Sort for more details!


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