Python Program - Merge Sort


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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


public class MyClass {
  // 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++) 
        {  
          System.out.print(Array[i] + " "); 
        }
        System.out.println(); 
    } 

  //test merge sort code
  public static void main(String[] args) {
    int[] MyArray = {10, 1, 23, 50, 4, 9, -4};
    int n = MyArray.length;
    System.out.println("Original Array");
    PrintArray(MyArray);

    mergesort(MyArray, 0, n-1);
    System.out.println("\nSorted Array");
    PrintArray(MyArray); 
  }
}

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 $$\mathcal{O}(LogN)$$ and merging process has time complexity $$\mathcal{O}(N)$$. Therefore, in all cases, the time complexity of merge sort is $$\mathcal{O}(NLogN)$$.




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