**Problem Statement:** Given an array of N integers, write a program to implement the Recursive Bubble Sort algorithm.

**Examples:**

Example 1:Input:N = 6, array[] = {13,46,24,52,20,9}Output:9,13,20,24,46,52Explanation:After sorting we get 9,13,20,24,46,52Example 2:Input:N = 5, array[] = {5,4,3,2,1}Output:1,2,3,4,5Explanation:After sorting we get 1,2,3,4,5

## Solution

** Disclaimer**:

*Don’t jump directly to the solution, try it out yourself first.*

**Solution:**

We have already solved this problem using the iterative method. To learn that approach, refer to the article: Bubble Sort Algorithm. In this article, we will solve this problem using recursion instead of using the loop.

**Approach**:

In the iterative method, we usually select a range(using a loop), and for each range, we repeatedly swap(using another loop) the adjacent elements(if arr[i] > arr[i+1]) until the maximum element in that range reaches the end.

The flow of the algorithm goes like this: in the first iteration, we select the range 0 to n-1 and swap adjacent elements(if arr[i] > arr[i+1]) until the maximum element reaches the (n-1)th index. Similarly, in the second iteration, the second maximum element reaches the (n-2)th index. So, the sorting basically occurs in the backward direction. After (n-1) such iterations we get the sorted array.

Now, in the recursive approach, we will just select the range recursively instead of using any loop. This is the only change we will do the recursive bubble sort algorithm and the rest of the part will be completely the same as it was in the case of iterative bubble sort.

The approach will be the following:

- First, call the recursive function with the given array and the range of n(size of the array).
- Inside that recursive function, repeatedly swap 2 adjacent elements if arr[j] > arr[j+1].

Here, the maximum element of the unsorted array reaches the end of the unsorted array after each recursive call. - Each time after step 2, call the recursion again decreasing the range by 1.
- The recursion will continue until the range(i.e. the size) of the array is reduced to 1.
**Base Case:**if(n == 1) return;

**Dry run:**

**Recursion 1: bubble_sort(arr, 6):**

**Recursion 2: bubble_sort(arr, 5):**

**Recursion 3: bubble_sort(arr, 4):**

**Recursion 4: bubble_sort(arr, 3):**

**Recursion 5: bubble_sort(arr, 2):**

**Recursion 6: bubble_sort(arr, 1):**Now, the recursion will hit the base case and it will return from this step.

**Code:**

## C++ Code

```
#include <bits/stdc++.h>
using namespace std;
void bubble_sort(int arr[], int n) {
// Base Case: range == 1.
if (n == 1) return;
for (int j = 0; j <= n - 2; j++) {
if (arr[j] > arr[j + 1]) {
int temp = arr[j + 1];
arr[j + 1] = arr[j];
arr[j] = temp;
}
}
//Range reduced after recursion:
bubble_sort(arr, n - 1);
}
int main()
{
int arr[] = {13, 46, 24, 52, 20, 9};
int n = sizeof(arr) / sizeof(arr[0]);
cout << "Before Using Bubble Sort: " << endl;
for (int i = 0; i < n; i++)
{
cout << arr[i] << " ";
}
cout << endl;
bubble_sort(arr, n);
cout << "After Using bubble sort: " << "\n";
for (int i = 0; i < n; i++) {
cout << arr[i] << " ";
}
cout << "\n";
return 0;
}
```

**Output: **Before Using Bubble Sort:

13 46 24 52 20 9

After Using bubble sort:

9 13 20 24 46 52

**Time Complexity: **O(N^{2}), (where N = size of the array), for the worst, and average cases.

**Reason:** If we carefully observe, we can notice that the recursion call, is occurring for n times, and for each recursion call, the loop j runs from 0 to n-2. For, the range of size n, the inner loop runs n-1 times, for the range of size n-1, the inner loop runs n-2 times, and so on. So, the total steps will be approximately the following: (n-1) + (n-2) + (n-3) + ……..+ 3 + 2 + 1. The summation is approximately the sum of the first n natural numbers i.e. (n*(n+1))/2. The precise time complexity will be O(n^{2}/2 + n/2). Previously, we have learned that we can ignore the lower values as well as the constant coefficients. So, the time complexity is O(n^{2}). Here the value of n is N i.e. the size of the array.

**Space Complexity: **O(N) auxiliary stack space.

## Java Code

```
import java.util.*;
public class tUf {
static void bubble_sort(int[] arr, int n) {
//Base case: range == 1.
if (n == 1) return;
for (int j = 0; j <= n - 2; j++) {
if (arr[j] > arr[j + 1]) {
int temp = arr[j];
arr[j] = arr[j + 1];
arr[j + 1] = temp;
}
}
//Range reduced after recursion:
bubble_sort(arr, n - 1);
}
public static void main(String args[]) {
int arr[] = {13, 46, 24, 52, 20, 9};
int n = arr.length;
System.out.println("Before Using Bubble Sort: ");
for (int i = 0; i < n; i++) {
System.out.print(arr[i] + " ");
}
System.out.println();
bubble_sort(arr, n);
System.out.println("After bubble sort: ");
for (int i = 0; i < n; i++) {
System.out.print(arr[i] + " ");
}
System.out.println();
}
}
```

**Output: **Before Using Bubble Sort:

13 46 24 52 20 9

After Using bubble sort:

9 13 20 24 46 52

**Time Complexity: **O(N^{2}), (where N = size of the array), for the worst, and average cases.

**Reason:** If we carefully observe, we can notice that the recursion call, is occurring for n times, and for each recursion call, the loop j runs from 0 to n-2. For, the range of size n, the inner loop runs n-1 times, for the range of size n-1, the inner loop runs n-2 times, and so on. So, the total steps will be approximately the following: (n-1) + (n-2) + (n-3) + ……..+ 3 + 2 + 1. The summation is approximately the sum of the first n natural numbers i.e. (n*(n+1))/2. The precise time complexity will be O(n^{2}/2 + n/2). Previously, we have learned that we can ignore the lower values as well as the constant coefficients. So, the time complexity is O(n^{2}). Here the value of n is N i.e. the size of the array.

**Space Complexity: **O(N) auxiliary stack space.

**Optimized approach (Reducing time complexity for the best case):**

The best case occurs if the given array is already sorted. We can reduce the time complexity to O(N) by just adding a small check inside the recursive function.

- We will check in the first recursion call if any swap is taking place. If the array is already sorted no swap will occur and we will return from the recursion call.
- Thus the number of recursions will be just 1. And our overall time complexity will be O(N).

**Code**:

## C++ Code

```
#include <bits/stdc++.h>
using namespace std;
void bubble_sort(int arr[], int n) {
// Base Case: range == 1.
if (n == 1) return;
int didSwap = 0;
for (int j = 0; j <= n - 2; j++) {
if (arr[j] > arr[j + 1]) {
int temp = arr[j + 1];
arr[j + 1] = arr[j];
arr[j] = temp;
didSwap = 1;
}
}
// if no swapping happens.
if (didSwap == 0) return;
//Range reduced after recursion:
bubble_sort(arr, n - 1);
}
int main()
{
int arr[] = {13, 46, 24, 52, 20, 9};
int n = sizeof(arr) / sizeof(arr[0]);
cout << "Before Using Bubble Sort: " << endl;
for (int i = 0; i < n; i++)
{
cout << arr[i] << " ";
}
cout << endl;
bubble_sort(arr, n);
cout << "After Using bubble sort: " << "\n";
for (int i = 0; i < n; i++) {
cout << arr[i] << " ";
}
cout << "\n";
return 0;
}
```

**Output: **Before Using Bubble Sort:

13 46 24 52 20 9

After Using bubble sort:

9 13 20 24 46 52

**Time Complexity: **O(N^{2}) for the worst and average cases and O(N) for the best case. Here, N = size of the array.

**Space Complexity: **O(N) auxiliary stack space.

## Java Code

```
import java.util.*;
public class tUf {
static void bubble_sort(int[] arr, int n) {
//Base case: range == 1.
if (n == 1) return;
int didSwap = 0;
for (int j = 0; j <= n - 2; j++) {
if (arr[j] > arr[j + 1]) {
int temp = arr[j];
arr[j] = arr[j + 1];
arr[j + 1] = temp;
didSwap = 1;
}
}
// if no swapping happens.
if (didSwap == 0) return;
//Range reduced after recursion:
bubble_sort(arr, n - 1);
}
public static void main(String args[]) {
int arr[] = {13, 46, 24, 52, 20, 9};
int n = arr.length;
System.out.println("Before Using Bubble Sort: ");
for (int i = 0; i < n; i++) {
System.out.print(arr[i] + " ");
}
System.out.println();
bubble_sort(arr, n);
System.out.println("After bubble sort: ");
for (int i = 0; i < n; i++) {
System.out.print(arr[i] + " ");
}
System.out.println();
}
}
```

**Output: **Before Using Bubble Sort:

13 46 24 52 20 9

After Using bubble sort:

9 13 20 24 46 52

**Time Complexity: **O(N^{2}) for the worst and average cases and O(N) for the best case. Here, N = size of the array.

**Space Complexity: **O(N) auxiliary stack space.

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