# Find a permutation that causes worst case of Merge Sort

Given a set of elements, find which permutation of these elements would result in worst case of Merge Sort?

Asymptotically, merge sort always takes ?(n Log n) time, but the cases that require more comparisons generally take more time in practice. We basically need to find a permutation of input elements that would lead to maximum number of comparisons when sorted using a typical Merge Sort algorithm.

Example:

```Consider the below set of elements
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16}

Below permutation of the set causes 153
comparisons.
{1, 9, 5, 13, 3, 11, 7, 15, 2, 10, 6,
14, 4, 12, 8, 16}

And an already sorted permutation causes
30 comparisons.

See this for a program that counts
comparisons and shows above results.```

Now how to get worst case input for merge sort for an input set?

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Lets us try to build the array in bottom up manner

Let the sorted array be {1,2,3,4,5,6,7,8}.

In order to generate the worst case of merge sort, the merge operation that resulted in above sorted array should result in maximum comparisons. In order to do so, the left and right sub-array involved in merge operation should store alternate elements of sorted array. i.e. left sub-array should be {1,3,5,7} and right sub-array should be {2,4,6,8}. Now every element of array will be compared at-least once and that will result in maximum comparisons. We apply the same logic for left and right sub-array as well. For array {1,3,5,7}, the worst case will be when its left and right sub-array are {1,5} and {3,7} respectively and for array {2,4,6,8} the worst case will occur for {2,4} and {6,8}.

Complete Algorithm –

GenerateWorstCase(arr[])

1. 1. Create two auxiliary arrays left and right and store alternate array elements in them.
2. Call GenerateWorstCase for left subarray: GenerateWorstCase (left)
3. Call GenerateWorstCase for right subarray: GenerateWorstCase (right)
4. Copy all elements of left and right subarrays back to original array.

Below is implementation of the idea

## C/C++

 `// C/C++ program to generate Worst Case of Merge Sort ` `#include ` `#include ` ` `  `// Function to print an array ` `void` `printArray(``int` `A[], ``int` `size) ` `{ ` `    ``for` `(``int` `i = 0; i < size; i++) ` `        ``printf``(``"%d "``, A[i]); ` ` `  `    ``printf``(``"\n"``); ` `} ` ` `  `// Function to join left and right subarray ` `int` `join(``int` `arr[], ``int` `left[], ``int` `right[], ` `          ``int` `l, ``int` `m, ``int` `r) ` `{ ` `    ``int` `i; ``// Used in second loop ` `    ``for` `(i = 0; i <= m - l; i++) ` `        ``arr[i] = left[i]; ` ` `  `    ``for` `(``int` `j = 0; j < r - m; j++) ` `        ``arr[i + j] = right[j]; ` `} ` ` `  `// Function to store alternate elemets in left ` `// and right subarray ` `int` `split(``int` `arr[], ``int` `left[], ``int` `right[], ` `          ``int` `l, ``int` `m, ``int` `r) ` `{ ` `    ``for` `(``int` `i = 0; i <= m - l; i++) ` `        ``left[i] = arr[i * 2]; ` ` `  `    ``for` `(``int` `i = 0; i < r - m; i++) ` `        ``right[i] = arr[i * 2 + 1]; ` `} ` ` `  `// Function to generate Worst Case of Merge Sort ` `int` `generateWorstCase(``int` `arr[], ``int` `l, ``int` `r) ` `{ ` `    ``if` `(l < r) ` `    ``{ ` `        ``int` `m = l + (r - l) / 2; ` ` `  `        ``// create two auxiliary arrays ` `        ``int` `left[m - l + 1]; ` `        ``int` `right[r - m]; ` ` `  `        ``// Store alternate array elements in left ` `        ``// and right subarray ` `        ``split(arr, left, right, l, m, r); ` ` `  `        ``// Recurse first and second halves ` `        ``generateWorstCase(left, l, m); ` `        ``generateWorstCase(right, m + 1, r); ` ` `  `        ``// join left and right subarray ` `        ``join(arr, left, right, l, m, r); ` `    ``} ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``// Sorted array ` `    ``int` `arr[] = { 1, 2, 3, 4, 5, 6, 7, 8, 9, ` `                 ``10, 11, 12, 13, 14, 15, 16 }; ` `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr); ` ` `  `    ``printf``(``"Sorted array is \n"``); ` `    ``printArray(arr, n); ` ` `  `    ``// generate Worst Case of Merge Sort ` `    ``generateWorstCase(arr, 0, n - 1); ` ` `  `    ``printf``(``"\nInput array that will result in "` `             ``"worst case of merge sort is \n"``); ` `    ``printArray(arr, n); ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java program to generate Worst Case of Merge Sort ` ` `  `import` `java.util.Arrays; ` ` `  `class` `GFG  ` `{ ` `    ``// Function to join left and right subarray ` `    ``static` `void` `join(``int` `arr[], ``int` `left[], ``int` `right[], ` `                    ``int` `l, ``int` `m, ``int` `r) ` `    ``{ ` `        ``int` `i; ` `        ``for` `(i = ``0``; i <= m - l; i++) ` `            ``arr[i] = left[i]; ` `  `  `        ``for` `(``int` `j = ``0``; j < r - m; j++) ` `            ``arr[i + j] = right[j]; ` `    ``} ` `  `  `    ``// Function to store alternate elemets in left ` `    ``// and right subarray ` `    ``static` `void` `split(``int` `arr[], ``int` `left[], ``int` `right[], ` `                     ``int` `l, ``int` `m, ``int` `r) ` `    ``{ ` `        ``for` `(``int` `i = ``0``; i <= m - l; i++) ` `            ``left[i] = arr[i * ``2``]; ` `  `  `        ``for` `(``int` `i = ``0``; i < r - m; i++) ` `            ``right[i] = arr[i * ``2` `+ ``1``]; ` `    ``} ` `     `  `    ``// Function to generate Worst Case of Merge Sort ` `    ``static` `void` `generateWorstCase(``int` `arr[], ``int` `l, ``int` `r) ` `    ``{ ` `        ``if` `(l < r) ` `        ``{ ` `            ``int` `m = l + (r - l) / ``2``; ` `  `  `            ``// create two auxiliary arrays ` `            ``int``[] left = ``new` `int``[m - l + ``1``]; ` `            ``int``[] right = ``new` `int``[r - m]; ` `  `  `            ``// Store alternate array elements in left ` `            ``// and right subarray ` `            ``split(arr, left, right, l, m, r); ` `  `  `            ``// Recurse first and second halves ` `            ``generateWorstCase(left, l, m); ` `            ``generateWorstCase(right, m + ``1``, r); ` `  `  `            ``// join left and right subarray ` `            ``join(arr, left, right, l, m, r); ` `        ``} ` `    ``} ` `     `  `    ``// driver program ` `    ``public` `static` `void` `main (String[] args)  ` `    ``{ ` `        ``// sorted array ` `        ``int` `arr[] = { ``1``, ``2``, ``3``, ``4``, ``5``, ``6``, ``7``, ``8``, ``9``, ` `                      ``10``, ``11``, ``12``, ``13``, ``14``, ``15``, ``16` `}; ` `        ``int` `n = arr.length; ` `        ``System.out.println(``"Sorted array is"``); ` `        ``System.out.println(Arrays.toString(arr)); ` `         `  `        ``// generate Worst Case of Merge Sort ` `        ``generateWorstCase(arr, ``0``, n - ``1``); ` `  `  `        ``System.out.println(``"\nInput array that will result in \n"``+ ` `             ``"worst case of merge sort is \n"``); ` `     `  `        ``System.out.println(Arrays.toString(arr)); ` `    ``} ` `} ` ` `  `// Contributed by Pramod Kumar `

## C#

 `// C# program to generate Worst Case of ` `// Merge Sort ` `using` `System; ` ` `  `class` `GFG { ` `     `  `    ``// Function to join left and right subarray ` `    ``static` `void` `join``(``int` `[]arr, ``int` `[]left,  ` `              ``int` `[]right, ``int` `l, ``int` `m, ``int` `r) ` `    ``{ ` `        ``int` `i; ` `        ``for` `(i = 0; i <= m - l; i++) ` `            ``arr[i] = left[i]; ` ` `  `        ``for` `(``int` `j = 0; j < r - m; j++) ` `            ``arr[i + j] = right[j]; ` `    ``} ` ` `  `    ``// Function to store alternate elemets in ` `    ``// left and right subarray ` `    ``static` `void` `split(``int` `[]arr, ``int` `[]left, ` `            ``int` `[]right, ``int` `l, ``int` `m, ``int` `r) ` `    ``{ ` `        ``for` `(``int` `i = 0; i <= m - l; i++) ` `            ``left[i] = arr[i * 2]; ` ` `  `        ``for` `(``int` `i = 0; i < r - m; i++) ` `            ``right[i] = arr[i * 2 + 1]; ` `    ``} ` `     `  `    ``// Function to generate Worst Case of  ` `    ``// Merge Sort ` `    ``static` `void` `generateWorstCase(``int` `[]arr,  ` `                                ``int` `l, ``int` `r) ` `    ``{ ` `        ``if` `(l < r) ` `        ``{ ` `            ``int` `m = l + (r - l) / 2; ` ` `  `            ``// create two auxiliary arrays ` `            ``int``[] left = ``new` `int``[m - l + 1]; ` `            ``int``[] right = ``new` `int``[r - m]; ` ` `  `            ``// Store alternate array elements ` `            ``// in left and right subarray ` `            ``split(arr, left, right, l, m, r); ` ` `  `            ``// Recurse first and second halves ` `            ``generateWorstCase(left, l, m); ` `            ``generateWorstCase(right, m + 1, r); ` ` `  `            ``// join left and right subarray ` `            ``join``(arr, left, right, l, m, r); ` `        ``} ` `    ``} ` `     `  `    ``// driver program ` `    ``public` `static` `void` `Main ()  ` `    ``{ ` `         `  `        ``// sorted array ` `        ``int` `[]arr = { 1, 2, 3, 4, 5, 6, 7, 8, 9, ` `                    ``10, 11, 12, 13, 14, 15, 16 }; ` `                     `  `        ``int` `n = arr.Length; ` `        ``Console.Write(``"Sorted array is\n"``); ` `         `  `        ``for``(``int` `i = 0; i < n; i++) ` `            ``Console.Write(arr[i] + ``" "``); ` `         `  `        ``// generate Worst Case of Merge Sort ` `        ``generateWorstCase(arr, 0, n - 1); ` ` `  `        ``Console.Write(``"\nInput array that will "` `                  ``+ ``"result in \n worst case of"` `                         ``+ ``" merge sort is \n"``); ` `     `  `        ``for``(``int` `i = 0; i < n; i++) ` `            ``Console.Write(arr[i] + ``" "``); ` `    ``} ` `} ` ` `  `// This code is contributed by Smitha  `

Output:

```Sorted array is
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Input array that will result in worst
case of merge sort is
1 9 5 13 3 11 7 15 2 10 6 14 4 12 8 16
```

References – Stack Overflow

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