Efficiently merging two sorted arrays with O(1) extra space and O(NlogN + MlogM)

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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 <iostream>
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 contain the numbers
    // which should be there in the result
    // Sort the array individually 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 contain the numbers 
        // which should be there in the result 
        // Sort the array individually 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 contain the numbers 
    # which should be there in the result 
    # Sort the array individually 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 contain the numbers 
        // which should be there in the result 
        // Sort the array individually 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

<script>
    // Javascript implementation of the approach 
     
    // Reducing the gap by a factor of 2 
    function nextGap(gap) 
    { 
        if (gap <= 1) 
            return 0; 
        return (parseInt(gap / 2, 10) + (gap % 2)); 
    } 
       
    // Function to merge two sorted 
    // arrays with O(1) extra space 
    function mergeTwoSortedArray(arr1, arr2, n, m) 
    { 
        let x = Math.min(n, m); 
       
        // Form both arrays to be bitonic 
        for (let i = 0; i < x; i++) 
        { 
            if (arr1[n - i - 1] > arr2[i]) 
            {
                let temp = arr1[n - i - 1];
                arr1[n - i - 1] = arr2[i];
                arr2[i] = temp;
            }
        } 
       
        // Now both the arrays contain the numbers 
        // which should be there in the result 
        // Sort the array individually by inplace algo 
        for (let gap = nextGap(n); gap > 0; 
            gap = nextGap(gap))
        { 
       
            // Comparing elements in the first array 
            for (let i = 0; i + gap < n; i++) 
                if (arr1[i] > arr1[i + gap])
                {
                    let temp = arr1[i];
                    arr1[i] = arr1[i + gap];
                    arr1[i + gap] = temp;
                }
        } 
       
        // Sort the second array 
        for (let gap = nextGap(m); gap > 0; 
            gap = nextGap(gap)) 
        { 
       
            // Comparing elements in the second array 
            for (let i = 0; i + gap < m; i++) 
                if (arr2[i] > arr2[i + gap])
                {
                    let temp = arr2[i];
                    arr2[i] = arr2[i + gap];
                    arr2[i + gap] = temp;
                }
        } 
        for (let i = 0; i < n; i++) 
            document.write(arr1[i] + " "); 
        for (let j = 0; j < m; j++) 
            document.write(arr2[j] + " "); 
    }
     
    let arr1 = [ 1, 5, 9, 10, 15, 20 ]; 
    let n = arr1.length; 
    let arr2 = [ 2, 3, 8, 13 ]; 
    let m = arr2.length; 
 
    mergeTwoSortedArray(arr1, arr2, n, m); 
         
</script>

Output:

1 2 3 5 8 9 10 13 15 20

Time Complexity: O(Nlog(N) + Mlog(M))
Space Complexity: O(1) as no extra space has been used.

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