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Merge Sort with O(1) extra space merge and O(n lg n) time
• Difficulty Level : Expert
• Last Updated : 18 Dec, 2020

We have discussed Merge sort. How to modify the algorithm so that merge works in O(1) extra space and algorithm still works in O(n Log n) time. We may assume that the input values are integers only.

Examples:

```Input : 5 4 3 2 1
Output : 1 2 3 4 5

Input : 999 612 589 856 56 945 243
Output : 56 243 589 612 856 945 999```

For integer types, merge sort can be made inplace using some mathematics trick of modulus and division. That means storing two elements value at one index and can be extracted using modulus and division.

First we have to find a value greater than all the elements of the array. Now we can store the original value as modulus and the second value as division. Suppose we want to store arr[i] and arr[j] both at index i(means in arr[i]). First we have to find a ‘maxval’ greater than both arr[i] and arr[j]. Now we can store as arr[i] = arr[i] + arr[j]*maxval. Now arr[i]%maxval will give the original value of arr[i] and arr[i]/maxval will give the value of arr[j]. So below is the implementation on merge sort.

C++

 `// C++ program to sort an array using merge sort such` `// that merge operation takes O(1) extra space.` `#include ` `using` `namespace` `std;` `void` `merge(``int` `arr[], ``int` `beg, ``int` `mid, ``int` `end, ``int` `maxele)` `{` `    ``int` `i = beg;` `    ``int` `j = mid + 1;` `    ``int` `k = beg;` `    ``while` `(i <= mid && j <= end) {` `        ``if` `(arr[i] % maxele <= arr[j] % maxele) {` `            ``arr[k] = arr[k] + (arr[i] % maxele) * maxele;` `            ``k++;` `            ``i++;` `        ``}` `        ``else` `{` `            ``arr[k] = arr[k] + (arr[j] % maxele) * maxele;` `            ``k++;` `            ``j++;` `        ``}` `    ``}` `    ``while` `(i <= mid) {` `        ``arr[k] = arr[k] + (arr[i] % maxele) * maxele;` `        ``k++;` `        ``i++;` `    ``}` `    ``while` `(j <= end) {` `        ``arr[k] = arr[k] + (arr[j] % maxele) * maxele;` `        ``k++;` `        ``j++;` `    ``}`   `    ``// Obtaining actual values` `    ``for` `(``int` `i = beg; i <= end; i++)` `        ``arr[i] = arr[i] / maxele;` `}`   `// Recursive merge sort with extra parameter, naxele` `void` `mergeSortRec(``int` `arr[], ``int` `beg, ``int` `end, ``int` `maxele)` `{` `    ``if` `(beg < end) {` `        ``int` `mid = (beg + end) / 2;` `        ``mergeSortRec(arr, beg, mid, maxele);` `        ``mergeSortRec(arr, mid + 1, end, maxele);` `        ``merge(arr, beg, mid, end, maxele);` `    ``}` `}`   `// This functions finds max element and calls recursive` `// merge sort.` `void` `mergeSort(``int` `arr[], ``int` `n)` `{` `   ``int` `maxele = *max_element(arr, arr+n) + 1;` `   ``mergeSortRec(arr, 0, n-1, maxele);` `}`   `int` `main()` `{` `    ``int` `arr[] = { 999, 612, 589, 856, 56, 945, 243 };` `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr[0]);` `    `  `    ``mergeSort(arr, n);`   `    ``cout << ``"Sorted array \n"``;` `    ``for` `(``int` `i = 0; i < n; i++)` `        ``cout << arr[i] << ``" "``;` `    ``return` `0;` `}`

Java

 `// Java program to sort an array ` `// using merge sort such that ` `// merge operation takes O(1) ` `// extra space.` `import` `java.util.Arrays;`   `class` `GFG ` `{`   `    ``static` `void` `merge(``int``[] arr, ``int` `beg,` `                        ``int` `mid, ``int` `end,` `                        ``int` `maxele) ` `    ``{` `        ``int` `i = beg;` `        ``int` `j = mid + ``1``;` `        ``int` `k = beg;` `        ``while` `(i <= mid && j <= end) ` `        ``{` `            ``if` `(arr[i] % maxele <=` `                ``arr[j] % maxele)` `            ``{` `                ``arr[k] = arr[k] + (arr[i]` `                        ``% maxele) * maxele;` `                ``k++;` `                ``i++;` `            ``} ` `            ``else` `            ``{` `                ``arr[k] = arr[k] + ` `                        ``(arr[j] % maxele)` `                        ``* maxele;` `                ``k++;` `                ``j++;` `            ``}` `        ``}` `        ``while` `(i <= mid) ` `        ``{` `            ``arr[k] = arr[k] + (arr[i]` `                    ``% maxele) * maxele;` `            ``k++;` `            ``i++;` `        ``}` `        ``while` `(j <= end) ` `        ``{` `            ``arr[k] = arr[k] + (arr[j]` `                    ``% maxele) * maxele;` `            ``k++;` `            ``j++;` `        ``}`   `        ``// Obtaining actual values ` `        ``for` `(i = beg; i <= end; i++)` `        ``{` `            ``arr[i] = arr[i] / maxele;` `        ``}` `    ``}`   `    ``// Recursive merge sort ` `    ``// with extra parameter, naxele ` `    ``static` `void` `mergeSortRec(``int``[] arr, ``int` `beg,` `            ``int` `end, ``int` `maxele) ` `    ``{` `        ``if` `(beg < end)` `        ``{` `            ``int` `mid = (beg + end) / ``2``;` `            ``mergeSortRec(arr, beg,` `                        ``mid, maxele);` `            ``mergeSortRec(arr, mid + ``1``,` `                        ``end, maxele);` `            ``merge(arr, beg, mid,` `                    ``end, maxele);` `        ``}` `    ``}`   `    ``// This functions finds ` `    ``// max element and calls ` `    ``// recursive merge sort. ` `    ``static` `void` `mergeSort(``int``[] arr, ``int` `n) ` `    ``{` `        ``int` `maxele = Arrays.stream(arr).max().getAsInt() + ``1``;` `        ``mergeSortRec(arr, ``0``, n - ``1``, maxele);` `    ``}`   `    ``// Driver code ` `    ``public` `static` `void` `main(String[] args)` `    ``{`   `        ``int``[] arr = {``999``, ``612``, ``589``,` `                    ``856``, ``56``, ``945``, ``243``};` `        ``int` `n = arr.length;`   `        ``mergeSort(arr, n);`   `        ``System.out.println(``"Sorted array "``);` `        ``for` `(``int` `i = ``0``; i < n; i++) ` `        ``{` `            ``System.out.print(arr[i] + ``" "``);` `        ``}` `    ``}` `} `   `// This code is contributed by 29AjayKumar`

Python3

 `# Python3 program to sort an array using` `# merge sort such that merge operation` `# takes O(1) extra space.` `def` `merge(arr, beg, mid, end, maxele):` `    `  `    ``i ``=` `beg` `    ``j ``=` `mid ``+` `1` `    ``k ``=` `beg` `    `  `    ``while` `(i <``=` `mid ``and` `j <``=` `end):` `        ``if` `(arr[i] ``%` `maxele <``=` `arr[j] ``%` `maxele):` `            ``arr[k] ``=` `arr[k] ``+` `(arr[i] ``%` `                    ``maxele) ``*` `maxele` `            ``k ``+``=` `1` `            ``i ``+``=` `1` `        ``else``:` `            ``arr[k] ``=` `arr[k] ``+` `(arr[j] ``%` `                    ``maxele) ``*` `maxele` `            ``k ``+``=` `1` `            ``j ``+``=` `1` `            `  `    ``while` `(i <``=` `mid):` `        ``arr[k] ``=` `arr[k] ``+` `(arr[i] ``%` `                ``maxele) ``*` `maxele` `        ``k ``+``=` `1` `        ``i ``+``=` `1` `    ``while` `(j <``=` `end):` `        ``arr[k] ``=` `arr[k] ``+` `(arr[j] ``%` `                ``maxele) ``*` `maxele` `        ``k ``+``=` `1` `        ``j ``+``=` `1`   `    ``# Obtaining actual values` `    ``for` `i ``in` `range``(beg, end ``+` `1``):` `        ``arr[i] ``=` `arr[i] ``/``/` `maxele`   `# Recursive merge sort with extra ` `# parameter, naxele` `def` `mergeSortRec(arr, beg, end, maxele):` `    `  `    ``if` `(beg < end):` `        ``mid ``=` `(beg ``+` `end) ``/``/` `2` `        ``mergeSortRec(arr, beg, mid, maxele)` `        ``mergeSortRec(arr, mid ``+` `1``, end, maxele)` `        ``merge(arr, beg, mid, end, maxele)`   `# This functions finds max element and ` `# calls recursive merge sort.` `def` `mergeSort(arr, n):` `    `  `   ``maxele ``=` `max``(arr) ``+` `1` `   ``mergeSortRec(arr, ``0``, n ``-` `1``, maxele)`   `# Driver Code` `if` `__name__ ``=``=` `'__main__'``:`   `    ``arr ``=` `[ ``999``, ``612``, ``589``, ``856``, ``56``, ``945``, ``243` `]` `    ``n ``=`  `len``(arr)` `    ``mergeSort(arr, n)`   `    ``print``(``"Sorted array"``)` `    `  `    ``for` `i ``in` `range``(n):` `        ``print``(arr[i], end ``=` `" "``)`   `# This code is contributed by mohit kumar 29`

C#

 `// C# program to sort an array ` `// using merge sort such that ` `// merge operation takes O(1)` `// extra space.` `using` `System;` `using` `System.Linq;`   `class` `GFG ` `{` `static` `void` `merge(``int` `[]arr, ``int` `beg, ` `                  ``int` `mid, ``int` `end, ` `                  ``int` `maxele)` `{` `    ``int` `i = beg;` `    ``int` `j = mid + 1;` `    ``int` `k = beg;` `    ``while` `(i <= mid && j <= end)` `    ``{` `        ``if` `(arr[i] %` `            ``maxele <= arr[j] % maxele)` `        ``{` `            ``arr[k] = arr[k] + (arr[i] % ` `                     ``maxele) * maxele;` `            ``k++;` `            ``i++;` `        ``}` `        ``else` `        ``{` `            ``arr[k] = arr[k] + ` `                    ``(arr[j] % maxele) * ` `                              ``maxele;` `            ``k++;` `            ``j++;` `        ``}` `    ``}` `    ``while` `(i <= mid)` `    ``{` `        ``arr[k] = arr[k] + (arr[i] % ` `                 ``maxele) * maxele;` `        ``k++;` `        ``i++;` `    ``}` `    ``while` `(j <= end)` `    ``{` `        ``arr[k] = arr[k] + (arr[j] % ` `                 ``maxele) * maxele;` `        ``k++;` `        ``j++;` `    ``}`   `    ``// Obtaining actual values` `    ``for` `( i = beg; i <= end; i++)` `        ``arr[i] = arr[i] / maxele;` `}`   `// Recursive merge sort ` `// with extra parameter, naxele` `static` `void` `mergeSortRec(``int` `[]arr, ``int` `beg,` `                         ``int` `end, ``int` `maxele)` `{` `    ``if` `(beg < end) ` `    ``{` `        ``int` `mid = (beg + end) / 2;` `        ``mergeSortRec(arr, beg,` `                     ``mid, maxele);` `        ``mergeSortRec(arr, mid + 1, ` `                     ``end, maxele);` `        ``merge(arr, beg, mid, ` `              ``end, maxele);` `    ``}` `}`   `// This functions finds ` `// max element and calls ` `// recursive merge sort.` `static` `void` `mergeSort(``int` `[]arr, ``int` `n)` `{` `    ``int` `maxele = arr.Max() + 1;` `    ``mergeSortRec(arr, 0, n - 1, maxele);` `}`   `//Driver code` `public` `static` `void` `Main ()` `{` `    ``int` `[]arr = {999, 612, 589, ` `                 ``856, 56, 945, 243};` `    ``int` `n = arr.Length;`   `    ``mergeSort(arr, n);`   `    ``Console.WriteLine(``"Sorted array "``);` `    ``for` `(``int` `i = 0; i < n; i++)` `        ``Console.Write( arr[i] + ``" "``);` `}` `}`   `// This code is contributed` `// by inder_verma.`

Output:

```Sorted array
56 243 589 612 856 945 999```

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