Merge Sort

MergeSort is a Divide and Conquer algorithm. It divides input array in two halves, calls itself for the two halves and then merges the two sorted halves. The merg() function is used for merging two halves. The merge(arr, l, m, r) is 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 following C implementation for details.

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

The following diagram from wikipedia 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 in two halves till the size becomes 1. Once the size becomes 1, the merge processes comes into action and starts merging arrays back till the complete array is merged.


/* C program for merge sort */

/* Function to merge the two haves arr[l..m] and arr[m+1..r] of array arr[] */
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;
    j = 0;
    k = l;
    while (i < n1 && j < n2)
        if (L[i] <= R[j])
            arr[k] = L[i];
            arr[k] = R[j];

    /* Copy the remaining elements of L[], if there are any */
    while (i < n1)
        arr[k] = L[i];

    /* Copy the remaining elements of R[], if there are any */
    while (j < n2)
        arr[k] = R[j];

/* 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)
        int m = l+(r-l)/2; //Same as (l+r)/2, but avoids overflow for large l and h
        mergeSort(arr, l, m);
        mergeSort(arr, m+1, r);
        merge(arr, l, m, r);

/* Function to print an array */
void printArray(int A[], int size)
    int i;
    for (i=0; i < size; i++)
        printf("%d ", A[i]);

/* Driver program to test above functions */
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;


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

Applications of Merge Sort
1) Merge Sort is useful for sorting linked lists in O(nLogn) time. Other nlogn algorithms like Heap Sort, Quick Sort (average case nLogn) cannot be applied to linked lists.
2) Inversion Count Problem
3) Used in External Sorting

Other Sorting Algorithms on GeeksforGeeks/GeeksQuiz:

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

See GATE Corner for all information about GATE CS and Quiz Corner for all Quizzes on GeeksQuiz.

  • Archita

    Can yoi please give me the explanation of why in Merge sort we divide the list into 2 parts only ??? Why not 3 or 4 ?

  • Trish

    In the function “void mergesort” , we can define m= l+(h-l)/2 instead of m=l+h/2 because if l and r get last two indexes and that too of very high values,then we might have an overflow.

    • geeksquizgeeksquiz

      Trish: Thanks for inputs. We have updated the post. Keep it up!

      • Mohamed Fasil

        updation has compilation error for h variable… it shd be r and not h

        • GeeksforGeeks

          Thanks for pointing this out. We have changed h to r.

  • Vin

    You have to free R and L in the end of merge function

    • GeeksforGeeks

      Thanks for pointing this out. We have changed the code to use auto arrays instead of dynamically allocated arrays.

  • Ravi Prakash Giri

    What is the need of j<=n2 for copying data into array R[ ]? Cann't it be done by simply j<n2?