It divides the input array into two halves, calls itself for the two halves, and then 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. 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)
- Implementation:
/* 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]*/
i = 0; // Initial index of first subarray
j = 0; // Initial index of second subarray
k = l; // Initial index of merged subarray
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 h
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: 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)
Algorithmic Paradigm: Divide and Conquer
Sorting In Place: No in a typical implementation
Stable: Yes