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Merge Sort in C

The merge sort follows the principle of the divide and conquers algorithm. Firstly it divides the input of an array into two halves and then calls itself for the two halves, and it then merges the sorted two halves into one final array, which will be sorted. The merge function is used for merging the two halves.

For example, the merge (array, i, m, r) is the process that assumes that array [i...m] and array[m+1...r] are hence sorted and then merges the two sorted halves or sub-arrays into a final one. Merge sort is one f the popular sorting techniques of all with the worst case time complexity of O(n log n).

Divide and conquer technique:

Using the strategy of divide and conquer, a problem can be divided into many sub problems. Then the solution of each sub problem can be written and later can be combined in order to get the result of the main problem.

For example, consider an array of elements called ‘arr’ needs to be sorted. The array ‘arr’ will be divided into a subproblem which will then be sorted in the subsection of the array starting at the index ‘p’ and ending at ‘r’ denoted as ‘arr [ p… r]’.

  • Divide: if the index ‘q’ is considered to be halfway from ‘p to r’, then the array ‘arr’ can be split into the subarray, which can be divided as ‘arr [p … q]’ and ‘arr [q +1… r].
  • Conquer: while conquering, the programmer has to sort both of the sub-arrays present ‘arr [p … q]’ and ‘arr [q +1… r]. If the base case is not achieved, then it can be further divided into subarrays and sorted.
  • Combine: once the dividing and conquering steps are completed, combining the sorted subarrays is crucial. This is because both the subarrays ‘arr [p … q]’ and ‘arr [q +1… r] which have been sorted are not compiled for the results,, and it creates the sorted array from the original array ‘arr [ p… r]’.

Merge sort algorithm:

For the function “merge (array [], l, r)”:

If r > 1:

1. Find the middle point to divide the array into two halves:

       middle m = l+ (r-l)/2

2. Call mergeSort for the first half:

            Call mergeSort(arr, l, m)

3. All mergeSort for the 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)

E.g.:

#include<stdio.h>
 
void mergesort(int a[],int i,int j);
void merge(int a[],int i1,int j1,int i2,int j2);
 
int main()
{
int a[30],n,i;
printf("Enter no of elements:");
scanf("%d",&n);
printf("Enter array elements:");
for(i=0;i<n;i++)
scanf("%d",&a[i]);
mergesort(a,0,n-1);
printf("\nSorted array is :");
for(i=0;i<n;i++)
printf("%d ",a[i]);
return 0;
}
 
void mergesort(int a[],int i,int j)
{
int mid;
if(i<j)
{
mid=(i+j)/2;
mergesort(a,i,mid); //left recursion
mergesort(a,mid+1,j); //right recursion
merge(a,i,mid,mid+1,j); //merging of two sorted sub-arrays
}
}
 
void merge(int a[],int i1,int j1,int i2,int j2)
{
int temp[50]; //array used for merging
int i,j,k;
i=i1; //beginning of the first list
j=i2; //beginning of the second list
k=0;
while(i<=j1 && j<=j2) //while elements in both lists
{
if(a[i]<a[j])
temp[k++]=a[i++];
else
temp[k++]=a[j++];
}
while(i<=j1) //copy remaining elements of the first list
temp[k++]=a[i++];
while(j<=j2) //copy remaining elements of the second list
temp[k++]=a[j++];
//Transfer elements from temp[] back to a[]
for(i=i1,j=0;i<=j2;i++,j++)
a[i]=temp[j];
}

Output:

Enter no of elements: 6
Enter array elements: 5 4 1 3 7 3 
Sorted array is : 1 3 3 4 5 7 

Time complexity:

Since merge sort is a recursive algorithm, the time complexity can be written as

follows:

T (n) = 2T (n/2) + θ(n)

The time complexity of the merge sort algorithm is the same for all the three cases which include best, average, and worst cases = θ (n Log n)

This is because the merge sort always divides the array into two halves and takes linear time to merge two halves.

It has the auxiliary space of O (n).

Applications of the merge sort

  • Merge Sort helps sort linked lists in O(n log n) time. In the case of linked lists, the case is different mainly due to the difference in memory allocation of arrays and linked lists.
  • Unlike arrays, linked list nodes may not be adjacent in memory.
  • Unlike an array, in the linked list, we can insert items in the middle in O(1) extra space and O(1) time. Therefore, the merge operation of merge sort can be implemented without extra space for linked lists.
  • It can be used in the inversion count problem.
  • It can also be made use in external sorting.

Drawbacks of using merge sort:

  • It is slower when compared to other algorithms as it uses the dividing and conquering method. This is only suitable for smaller tasks.
  • Unlike, any other sorting techniques, Merge sort uses an additional space for the memory O (N) for the temporary array while it divides. This space is known as auxiliary space.
  • It always goes through the whole process of divide and conquers even though the array is sorted beforehand.



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