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Efficiently merging two sorted arrays with O(1) extra space and O(NlogN + MlogM)
• Difficulty Level : Easy
• Last Updated : 10 Jun, 2021

Given two sorted arrays, arr1[] and arr2[], the task is to merge them in O(Nlog(N) + Mlog(M)) time with O(1) extra space into a sorted array where N is the size of the first array arr1[] and M is the size of the second array arr2[].

Examples:

Input: arr1[] = {1, 5, 9, 10, 15, 20}, arr2[] = {2, 3, 8, 13}
Output: 1 2 3 5 8 9 10 13 15 20

Input: arr1[] = {4, 9, 15, 20}, arr2[] = {2, 6, 7, 13}
Output: 2 4 6 7 9 13 15 20

Approach: An approach has already been discussed in this article. In this article, an even more efficient approach will be discussed.

• Form two bitonic arrays by comparing the highest element of one array to the lowest element of the second array such that both arrays contain only those elements which are to be there after the sorting of both the arrays.
• Now, sort both the arrays separately.

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach` `#include ``using` `namespace` `std;` `// Reducing the gap by a factor of 2``int` `nextGap(``int` `gap)``{``    ``if` `(gap <= 1)``        ``return` `0;``    ``return` `(gap / 2) + (gap % 2);``}` `// Function to merge two sorted``// arrays with O(1) extra space``int` `mergeTwoSortedArray(``int``* arr1,``                        ``int``* arr2,``                        ``int` `n, ``int` `m)``{``    ``int` `x = min(n, m);` `    ``// Form both arrays to be bitonic``    ``for` `(``int` `i = 0; i < x; i++) {``        ``if` `(arr1[n - i - 1] > arr2[i])``            ``swap(arr1[n - i - 1], arr2[i]);``    ``}` `    ``// Now both the arrays conatin the numbers``    ``// which should be there in the result``    ``// Sort the array indiviually by inplace algo``    ``for` `(``int` `gap = nextGap(n); gap > 0;``         ``gap = nextGap(gap)) {` `        ``// Comparing elements in the first array``        ``for` `(``int` `i = 0; i + gap < n; i++)``            ``if` `(arr1[i] > arr1[i + gap])``                ``swap(arr1[i], arr1[i + gap]);``    ``}` `    ``// Sort the second array``    ``for` `(``int` `gap = nextGap(m); gap > 0;``         ``gap = nextGap(gap)) {` `        ``// Comparing elements in the second array``        ``for` `(``int` `i = 0; i + gap < m; i++)``            ``if` `(arr2[i] > arr2[i + gap])``                ``swap(arr2[i], arr2[i + gap]);``    ``}``    ``for` `(``int` `i = 0; i < n; i++)``        ``cout << arr1[i] << ``" "``;``    ``for` `(``int` `j = 0; j < m; j++)``        ``cout << arr2[j] << ``" "``;``}` `// Driver code``int` `main()``{``    ``int` `arr1[] = { 1, 5, 9, 10, 15, 20 };``    ``int` `n = ``sizeof``(arr1) / ``sizeof``(``int``);``    ``int` `arr2[] = { 2, 3, 8, 13 };``    ``int` `m = ``sizeof``(arr2) / ``sizeof``(``int``);` `    ``mergeTwoSortedArray(arr1, arr2, n, m);` `    ``return` `0;``}`

## Java

 `// Java implementation of the approach``class` `GFG``{``    ` `    ``// Reducing the gap by a factor of 2``    ``static` `int` `nextGap(``int` `gap)``    ``{``        ``if` `(gap <= ``1``)``            ``return` `0``;``        ``return` `(``int``)((gap / ``2``) + (gap % ``2``));``    ``}``    ` `    ``// Function to merge two sorted``    ``// arrays with O(1) extra space``    ``static` `void` `mergeTwoSortedArray(``int` `[]arr1,``                                    ``int` `[]arr2,``                                    ``int` `n, ``int` `m)``    ``{``        ``int` `x = Math.min(n, m);``    ` `        ``// Form both arrays to be bitonic``        ``for` `(``int` `i = ``0``; i < x; i++)``        ``{``            ``if` `(arr1[n - i - ``1``] > arr2[i])``            ``{``                ``// swap(arr1[n - i - 1], arr2[i]);``                ``int` `temp = arr1[n - i - ``1``];``                ``arr1[n - i - ``1``] = arr2[i];``                ``arr2[i] = temp;``            ``}``        ``}``    ` `        ``// Now both the arrays conatin the numbers``        ``// which should be there in the result``        ``// Sort the array indiviually by inplace algo``        ``for` `(``int` `gap = nextGap(n); gap > ``0``;``            ``gap = nextGap(gap))``        ``{``    ` `            ``// Comparing elements in the first array``            ``for` `(``int` `i = ``0``; i + gap < n; i++)``                ``if` `(arr1[i] > arr1[i + gap])``                ``{``                    ``// swap(arr1[i], arr1[i + gap]);``                    ``int` `temp = arr1[i];``                    ``arr1[i] = arr1[i + gap];``                    ``arr1[i + gap] = temp;``                ``}``        ``}``    ` `        ``// Sort the second array``        ``for` `(``int` `gap = nextGap(m); gap > ``0``;``            ``gap = nextGap(gap))``        ``{``    ` `            ``// Comparing elements in the second array``            ``for` `(``int` `i = ``0``; i + gap < m; i++)``                ``if` `(arr2[i] > arr2[i + gap])``                ``{``                    ``// swap(arr2[i], arr2[i + gap]);``                    ``int` `temp = arr2[i];``                    ``arr2[i] = arr2[i + gap];``                    ``arr2[i + gap] = temp;``                ``}``        ``}``        ``for` `(``int` `i = ``0``; i < n; i++)``            ``System.out.print(arr1[i] + ``" "``);``        ``for` `(``int` `j = ``0``; j < m; j++)``            ``System.out.print(arr2[j] + ``" "``);``    ``}``    ` `    ``// Driver code``    ``public` `static` `void` `main (String[] args)``    ``{``        ``int` `arr1[] = { ``1``, ``5``, ``9``, ``10``, ``15``, ``20` `};``        ``int` `n = arr1.length;``        ``int` `arr2[] = { ``2``, ``3``, ``8``, ``13` `};``        ``int` `m = arr2.length;``    ` `        ``mergeTwoSortedArray(arr1, arr2, n, m);``    ``}``}` `// This code is contributed by AnkitRai01`

## Python3

 `# Python3 implementation of the approach` `# Reducing the gap by a factor of 2``def` `nextGap(gap) :``    ``if` `(gap <``=` `1``) :``        ``return` `0``;``    ``res ``=` `(gap ``/``/` `2``) ``+` `(gap ``%` `2``);``    ``return` `res;` `# Function to merge two sorted``# arrays with O(1) extra space``def` `mergeTwoSortedArray(arr1, arr2, n, m) :` `    ``x ``=` `min``(n, m);` `    ``# Form both arrays to be bitonic``    ``for` `i ``in` `range``(x) :``        ``if` `(arr1[n ``-` `i ``-` `1``] > arr2[i]) :``            ``arr1[n ``-` `i ``-` `1``],arr2[i] ``=` `arr2[i], arr1[n``-` `i ``-` `1``];` `    ``# Now both the arrays conatin the numbers``    ``# which should be there in the result``    ``# Sort the array indiviually by inplace algo``    ``gap ``=` `nextGap(n);``    ``while` `gap > ``0` `:``        ` `        ``# Comparing elements in the first array``        ``i ``=` `0``;``        ` `        ``while` `i ``+` `gap < n :``            ` `            ``if` `(arr1[i] > arr1[i ``+` `gap]) :``                ``arr1[i], arr1[i ``+` `gap] ``=` `arr1[i ``+` `gap],arr1[i];``                ` `            ``i ``+``=` `1``;``            ` `        ``gap ``=` `nextGap(gap)` `    ``# Sort the second array``    ``gap ``=` `nextGap(m);``    ` `    ``while` `gap > ``0` `:``        ` `        ``# Comparing elements in the second array``        ``i ``=` `0``        ``while` `i ``+` `gap < m :``            ``if` `(arr2[i] > arr2[i ``+` `gap]) :``                ``arr2[i], arr2[i ``+` `gap] ``=` `arr2[i ``+` `gap], arr2[i];``                ` `            ``i ``+``=` `1``;``            ` `        ``gap ``=` `nextGap(gap)``            ` `    ``for` `i ``in` `range``(n) :``        ``print``(arr1[i], end ``=` `" "``);``        ` `    ``for` `j ``in` `range``(m) :``        ``print``(arr2[j], end ``=` `" "``);` `# Driver code``if` `__name__ ``=``=` `"__main__"` `:` `    ``arr1 ``=` `[ ``1``, ``5``, ``9``, ``10``, ``15``, ``20` `];``    ``n ``=` `len``(arr1);``    ``arr2 ``=` `[ ``2``, ``3``, ``8``, ``13` `];``    ``m ``=` `len``(arr2);` `    ``mergeTwoSortedArray(arr1, arr2, n, m);` `# This code is contributed by AnkitRai01`

## C#

 `// C# implementation of the approach``using` `System;` `class` `GFG``{``    ` `    ``// Reducing the gap by a factor of 2``    ``static` `int` `nextGap(``int` `gap)``    ``{``        ``if` `(gap <= 1)``            ``return` `0;``        ``return` `(``int``)((gap / 2) + (gap % 2));``    ``}``    ` `    ``// Function to merge two sorted``    ``// arrays with O(1) extra space``    ``static` `void` `mergeTwoSortedArray(``int` `[]arr1,``                                    ``int` `[]arr2,``                                    ``int` `n, ``int` `m)``    ``{``        ``int` `x = Math.Min(n, m);``    ` `        ``// Form both arrays to be bitonic``        ``for` `(``int` `i = 0; i < x; i++)``        ``{``            ``if` `(arr1[n - i - 1] > arr2[i])``            ``{``                ``int` `temp = arr1[n - i - 1];``                ``arr1[n - i - 1] = arr2[i];``                ``arr2[i] = temp;``            ``}``        ``}``    ` `        ``// Now both the arrays conatin the numbers``        ``// which should be there in the result``        ``// Sort the array indiviually by inplace algo``        ``for` `(``int` `gap = nextGap(n); gap > 0;``            ``gap = nextGap(gap))``        ``{``    ` `            ``// Comparing elements in the first array``            ``for` `(``int` `i = 0; i + gap < n; i++)``                ``if` `(arr1[i] > arr1[i + gap])``                ``{``                    ``int` `temp = arr1[i];``                    ``arr1[i] = arr1[i + gap];``                    ``arr1[i + gap] = temp;``                ``}``        ``}``    ` `        ``// Sort the second array``        ``for` `(``int` `gap = nextGap(m); gap > 0;``            ``gap = nextGap(gap))``        ``{``    ` `            ``// Comparing elements in the second array``            ``for` `(``int` `i = 0; i + gap < m; i++)``                ``if` `(arr2[i] > arr2[i + gap])``                ``{``                    ``int` `temp = arr2[i];``                    ``arr2[i] = arr2[i + gap];``                    ``arr2[i + gap] = temp;``                ``}``        ``}``        ``for` `(``int` `i = 0; i < n; i++)``            ``Console.Write(arr1[i] + ``" "``);``        ``for` `(``int` `j = 0; j < m; j++)``            ``Console.Write(arr2[j] + ``" "``);``    ``}``    ` `    ``// Driver code``    ``public` `static` `void` `Main()``    ``{``        ``int` `[]arr1 = { 1, 5, 9, 10, 15, 20 };``        ``int` `n = arr1.Length;``        ``int` `[]arr2 = { 2, 3, 8, 13 };``        ``int` `m = arr2.Length;``    ` `        ``mergeTwoSortedArray(arr1, arr2, n, m);``    ``}``}` `// This code is contributed by AnkitRai01`

## Javascript

 ``
Output:
`1 2 3 5 8 9 10 13 15 20`

Time Complexity: O(Nlog(N) + Mlog(M))

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