Python Program - Merge Sort
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]
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 split // 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 the 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); } }
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).
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