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 merge() 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.
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)
Example : The complete merge sort process for an example array {38, 27, 43, 3, 9, 82, 10}.
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. Sample Code:
/** * @author Abhinaw.Tripathi * */ public class MergeSortApp {
void merge(int arr[], int l, int m, int r) { // Find sizes of two subarrays to be merged int n1 = m - l + 1; int n2 = r - m; /* Create temp arrays */ int L[] = new int [n1]; int R[] = new int [n2]; /*Copy data to temp arrays*/ for (int i=0; i<n1; ++i) L[i] = arr[l + i]; for (int j=0; j<n2; ++j) R[j] = arr[m + 1+ j]; /* Merge the temp arrays */ // Initial indexes of first and second subarrays int i = 0, j = 0; // Initial index of merged subarry array int k = l; while (i < n1 && j < n2) { if (L[i] <= R[j]) { arr[k] = L[i]; i++; } else { arr[k] = R[j]; j++; } k++; } /* Copy remaining elements of L[] if any */ while (i < n1) { arr[k] = L[i]; i++; k++; } /* Copy remaining elements of L[] if any */ while (j < n2) { arr[k] = R[j]; j++; k++; } } // Main function that sorts arr[l..r] using // merge() void sort(int arr[], int l, int r) { if (l < r) { // Find the middle point int m = (l+r)/2; // Sort first and second halves sort(arr, l, m); sort(arr , m+1, r); // Merge the sorted halves merge(arr, l, m, r); } } public void printArray(int arr[]) { for(int i=0;i<arr.length;i++) { System.out.println(arr[i] + " "); } System.out.println(" "); } public static void main(String[] args) { int arr[] = {12, 11, 13, 5, 6, 7}; MergeSortApp ob = new MergeSortApp(); System.out.println("Given Array"); ob.printArray(arr); ob.sort(arr, 0, arr.length-1); System.out.println("\nSorted array"); ob.printArray(arr); }
}
Output:
Given Array 12 11 13 5 6 7 Sorted array 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