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Merge sort is a divide and conquer algorithm. It is based on the idea of dividing the unsorted array into several sub-array until each sub-array consists of a single element and merging those sub-array in such a way that results into a sorted array. The process step of merge sort can be summarized as follows:

  • Divide: Divide the unsorted array into several sub-array until each sub-array contains only single element.
  • Merge: Merge the sub-arrays in such way that results into sorted array and merge until achieves the original array.
  • Merging technique: the first element of the two sub-arrays is considered and compared. For ascending order sorting, the element with smaller value is taken from the sub-array and becomes a new element of the sorted array. This process is repeated until both sub-array are emptied and the merged array becomes sorted array.

Example:

To understand the merge sort, lets consider an unsorted array [4, 9, -4] (right side array created after 11th process in the below diagram) and discuss each step taken to sort the array in ascending order.

At the first step, the array [4, 9, -4] is divided into two sub-array. The first sub-array contains [4, 9] and second sub-array contains [-4]. As the number of element in the first sub-array is greater than one, it is further divided into sub-arrays consisting of elements [4] and [9] respectively. As the number of elements in all sub-arrays is one, hence the further dividing of the array can not be done.

In the merging process, The sub-arrays formed in the last step are combined together using the process mentioned above to form a sorted array. First, [4] and [9] sub-arrays are merged together to form a sorted sub-array [4, 9]. Then [4, 9] and [-4] sub-arrays are merged together to form final sorted array [-4, 4, 9]

Merge Sort

Implementation of Merge Sort

using System;

class MyProgram {
    // function for merge sort - splits the array 
    // and call merge function to sort and merge the array
    // mergesort is a recursive function
    static void mergesort(int[] Array, int left, int right) 
    {
      if (left < right)
      { 
        int mid = left + (right - left)/2;
        mergesort(Array, left, mid);
        mergesort(Array, mid+1, right);
        merge(Array, left, mid, right);
      }
    }

    // merge function performs sort and merge operations
    // for mergesort function
    static void merge(int[] Array, int left, int mid, int right) 
    {
        // Create two temporary array to hold splitted 
        // elements of main array 
        int n1 = mid - left + 1; //no of elements in LeftArray
        int n2 = right - mid;     //no of elements in RightArray    
        int[] LeftArray = new int [n1]; 
        int[] RightArray = new int [n2];
        for(int i=0; i < n1; i++)
         { 
           LeftArray[i] = Array[left + i];
         }
        for(int i=0; i < n2; i++)
         { 
           RightArray[i] = Array[mid + i + 1];
         }

      // In below section x, y and z represents index number
      // of LeftArray, RightArray and Array respectively
      int x=0, y=0, z=left;
      while(x < n1 && y < n2)
      {
        if(LeftArray[x] < RightArray[y])
           { 
              Array[z] = LeftArray[x]; 
              x++; 
           }
        else
           { 
             Array[z] = RightArray[y];  
             y++; 
           }
        z++;
      }
      // Copying the remaining elements of LeftArray
      while(x < n1)
      { 
         Array[z] = LeftArray[x];  
         x++;  z++;
      }
      // Copying the remaining elements of RightArray
      while(y < n2)
      { 
        Array[z] = RightArray[y]; 
        y++;  z++; 
      }
  }

   // function to print array
   static void PrintArray(int[] Array) 
   { 
      int n = Array.Length; 
      for (int i=0; i<n; i++) 
      {  
        Console.Write(Array[i] + " "); 
      }
      Console.Write("\n"); 
   } 

  // test the code
  static void Main(string[] args) {
   int[] MyArray = {10, 1, 23, 50, 4, 9, -4};
   int n = MyArray.Length;
   Console.Write("Original Array\n");
   PrintArray(MyArray);

   mergesort(MyArray, 0, n-1);
   Console.Write("\nSorted Array\n");
   PrintArray(MyArray);  
  }
}

The above code will give the following output:

Original Array
10 1 23 50 4 9 -4 

Sorted Array
-4 1 4 9 10 23 50

Time Complexity:

In all cases (worst, average and best), merge sort always divides the array until all sub-arrays contains single element and takes linear time to merge those sub-arrays. Dividing process has time complexity Θ(logN) and merging process has time complexity Θ(N). Therefore, in all cases, the time complexity of merge sort is Θ(NlogN).