Bubble Sort - Algorithm, Visualization, Examples

Problem Statement❯

Given an array of integers, your task is to sort the array in ascending order using the Bubble Sort algorithm.

Bubble Sort works by repeatedly stepping through the list, comparing adjacent elements, and swapping them if they are in the wrong order. The largest unsorted element 'bubbles up' to its correct position after each pass.

Your goal is to return the sorted array after all necessary passes are completed.

Examples❯

Input Array	Sorted Output	Description
[5, 1, 4, 2, 8]	[1, 2, 4, 5, 8]	Multiple elements in random order; standard caseVisualization
[3, 2, 1]	[1, 2, 3]	Completely reverse sorted; maximum swaps neededVisualization
[1, 2, 3]	[1, 2, 3]	Already sorted; best case (0 swaps)Visualization
[7]	[7]	Single-element array is always sortedVisualization
[]	[]	Empty array; nothing to sort
[4, 2, 2, 8, 4]	[2, 2, 4, 4, 8]	Array with duplicate valuesVisualization
[5, -1, 0, -3]	[-3, -1, 0, 5]	Array with negative numbersVisualization

Visualization Player

Solution❯

Bubble Sort works by repeatedly swapping adjacent elements that are in the wrong order, causing larger values to 'bubble up' to the end of the array after each pass.

Understanding the Behavior with Different Cases

Let’s walk through how Bubble Sort behaves in different scenarios so you can understand what to expect:

Normal case: For an array like [5, 1, 4, 8, 2], Bubble Sort will make several passes, each time pushing the largest remaining unsorted element to its correct position. After the first pass, the largest value (8) moves to the end. The next pass focuses only on the unsorted portion [1, 4, 5, 2], and so on until the array is fully sorted.
Reverse sorted case: For an input like [3, 2, 1], the algorithm performs the maximum number of swaps. Each element has to move all the way to its correct position, making this the worst-case scenario for Bubble Sort in terms of efficiency.
Already sorted case: If the array is already in order, like [1, 2, 3], Bubble Sort can detect this and finish early without doing any swaps—especially when optimized. This is its best-case scenario and shows the value of checking if any swaps occurred during a pass.
Duplicates and repeated values: Bubble Sort handles duplicates naturally. For example, [4, 2, 2, 8, 4] gets sorted to [2, 2, 4, 4, 8] just like any other array. Duplicate values are compared and only moved when necessary.
Negative numbers: The algorithm works the same with negative numbers. For example, [5, -1, 0, -3] becomes [-3, -1, 0, 5] after sorting.
Single-element array: An array like [7] is already sorted. No swaps or comparisons are needed, so the algorithm exits immediately.
Empty array: If the input is an empty array [], there's nothing to sort, and the output is also []. Bubble Sort simply doesn’t enter the loop.

Algorithm Steps❯

Start from the first element in the array.
Compare the current element with the next element.
If the current element is greater than the next element, swap them.
Move to the next element and repeat steps 2 and 3 for the entire array.
After one complete pass, the largest element will be at the end.
Repeat the process for the remaining unsorted elements (excluding the last sorted ones).
Continue until the array is completely sorted.

Code

Python

Java

JavaScript

C++

def bubble_sort(arr):
    n = len(arr)
    # Traverse through all array elements
    for i in range(n):
        # Last i elements are already in place
        for j in range(0, n - i - 1):
            # Swap if the element found is greater than the next element
            if arr[j] > arr[j + 1]:
                arr[j], arr[j + 1] = arr[j + 1], arr[j]
if __name__ == '__main__':
    arr = [6, 3, 8, 2, 7, 4]
    bubble_sort(arr)
    print("Sorted array is:", arr)

Time Complexity❯

Case	Time Complexity	Explanation
Best Case	O(n)	In the best case, the array is already sorted. Bubble Sort makes one pass through the array without any swaps, and with an optimized version that checks for swaps, it terminates early. Hence, the time complexity is linear.
Average Case	O(n^2)	On average, Bubble Sort compares and possibly swaps each pair of elements in a nested loop. This results in roughly n*(n-1)/2 operations, leading to quadratic time complexity.
Worst Case	O(n^2)	In the worst case (when the array is sorted in reverse order), Bubble Sort performs the maximum number of comparisons and swaps, resulting in n*(n-1)/2 operations — hence, O(n^2).

Space Complexity❯

O(1)

Explanation: Bubble Sort is an in-place sorting algorithm. It requires only a constant amount of extra space for swapping elements — typically a temporary variable and loop counters.

Detailed Step by Step Example❯

Let us take the followign array, and apply Bubble Sort algorithm to sort the array.

{ "array": [6,3,8,2,7,4], "showIndices": false, "specialIndices": [] }

Pass 1

➜ Comparing 6 and 3

{ "array": [6,3,8,2,7,4], "showIndices": false, "highlightIndices": [0,1], "highlightIndicesGreen": [], "specialIndices": [] }

6 > 3.
Swap 6 and 3.
We need to arrange them in ascending order.

{ "array": [3,6,8,2,7,4], "showIndices": false, "highlightIndicesBlue": [0,1], "highlightIndicesGreen": [], "specialIndices": [] }

➜ Comparing 6 and 8

{ "array": [3,6,8,2,7,4], "showIndices": false, "highlightIndices": [1,2], "highlightIndicesGreen": [], "specialIndices": [] }

6 <= 8.
No swap needed.
They are already in ascending order.

➜ Comparing 8 and 2

{ "array": [3,6,8,2,7,4], "showIndices": false, "highlightIndices": [2,3], "highlightIndicesGreen": [], "specialIndices": [] }

8 > 2.
Swap 8 and 2.
We need to arrange them in ascending order.

{ "array": [3,6,2,8,7,4], "showIndices": false, "highlightIndicesBlue": [2,3], "highlightIndicesGreen": [], "specialIndices": [] }

➜ Comparing 8 and 7

{ "array": [3,6,2,8,7,4], "showIndices": false, "highlightIndices": [3,4], "highlightIndicesGreen": [], "specialIndices": [] }

8 > 7.
Swap 8 and 7.
We need to arrange them in ascending order.

{ "array": [3,6,2,7,8,4], "showIndices": false, "highlightIndicesBlue": [3,4], "highlightIndicesGreen": [], "specialIndices": [] }

➜ Comparing 8 and 4

{ "array": [3,6,2,7,8,4], "showIndices": false, "highlightIndices": [4,5], "highlightIndicesGreen": [], "specialIndices": [] }

8 > 4.
Swap 8 and 4.
We need to arrange them in ascending order.

{ "array": [3,6,2,7,4,8], "showIndices": false, "highlightIndicesBlue": [4,5], "highlightIndicesGreen": [], "specialIndices": [] }

8 is now at its correct position.

{ "array": [3,6,2,7,4,8], "showIndices": false, "highlightIndicesGreen": [5], "specialIndices": [] }

Pass 2

➜ Comparing 3 and 6

{ "array": [3,6,2,7,4,8], "showIndices": false, "highlightIndices": [0,1], "highlightIndicesGreen": [5], "specialIndices": [] }

3 <= 6.
No swap needed.
They are already in ascending order.

➜ Comparing 6 and 2

{ "array": [3,6,2,7,4,8], "showIndices": false, "highlightIndices": [1,2], "highlightIndicesGreen": [5], "specialIndices": [] }

6 > 2.
Swap 6 and 2.
We need to arrange them in ascending order.

{ "array": [3,2,6,7,4,8], "showIndices": false, "highlightIndicesBlue": [1,2], "highlightIndicesGreen": [5], "specialIndices": [] }

➜ Comparing 6 and 7

{ "array": [3,2,6,7,4,8], "showIndices": false, "highlightIndices": [2,3], "highlightIndicesGreen": [5], "specialIndices": [] }

6 <= 7.
No swap needed.
They are already in ascending order.

➜ Comparing 7 and 4

{ "array": [3,2,6,7,4,8], "showIndices": false, "highlightIndices": [3,4], "highlightIndicesGreen": [5], "specialIndices": [] }

7 > 4.
Swap 7 and 4.
We need to arrange them in ascending order.

{ "array": [3,2,6,4,7,8], "showIndices": false, "highlightIndicesBlue": [3,4], "highlightIndicesGreen": [5], "specialIndices": [] }

7 is now at its correct position.

{ "array": [3,2,6,4,7,8], "showIndices": false, "highlightIndicesGreen": [4,5], "specialIndices": [] }

Pass 3

➜ Comparing 3 and 2

{ "array": [3,2,6,4,7,8], "showIndices": false, "highlightIndices": [0,1], "highlightIndicesGreen": [4,5], "specialIndices": [] }

3 > 2.
Swap 3 and 2.
We need to arrange them in ascending order.

{ "array": [2,3,6,4,7,8], "showIndices": false, "highlightIndicesBlue": [0,1], "highlightIndicesGreen": [4,5], "specialIndices": [] }

➜ Comparing 3 and 6

{ "array": [2,3,6,4,7,8], "showIndices": false, "highlightIndices": [1,2], "highlightIndicesGreen": [4,5], "specialIndices": [] }

3 <= 6.
No swap needed.
They are already in ascending order.

➜ Comparing 6 and 4

{ "array": [2,3,6,4,7,8], "showIndices": false, "highlightIndices": [2,3], "highlightIndicesGreen": [4,5], "specialIndices": [] }

6 > 4.
Swap 6 and 4.
We need to arrange them in ascending order.

{ "array": [2,3,4,6,7,8], "showIndices": false, "highlightIndicesBlue": [2,3], "highlightIndicesGreen": [4,5], "specialIndices": [] }

6 is now at its correct position.

{ "array": [2,3,4,6,7,8], "showIndices": false, "highlightIndicesGreen": [3,4,5], "specialIndices": [] }

Pass 4

➜ Comparing 2 and 3

{ "array": [2,3,4,6,7,8], "showIndices": false, "highlightIndices": [0,1], "highlightIndicesGreen": [3,4,5], "specialIndices": [] }

2 <= 3.
No swap needed.
They are already in ascending order.

➜ Comparing 3 and 4

{ "array": [2,3,4,6,7,8], "showIndices": false, "highlightIndices": [1,2], "highlightIndicesGreen": [3,4,5], "specialIndices": [] }

3 <= 4.
No swap needed.
They are already in ascending order.

4 is now at its correct position.

{ "array": [2,3,4,6,7,8], "showIndices": false, "highlightIndicesGreen": [2,3,4,5], "specialIndices": [] }

Pass 5

➜ Comparing 2 and 3

{ "array": [2,3,4,6,7,8], "showIndices": false, "highlightIndices": [0,1], "highlightIndicesGreen": [2,3,4,5], "specialIndices": [] }

2 <= 3.
No swap needed.
They are already in ascending order.

3 is now at its correct position.

{ "array": [2,3,4,6,7,8], "showIndices": false, "highlightIndicesGreen": [1,2,3,4,5], "specialIndices": [] }

Pass 6

2 is now at its correct position.

{ "array": [2,3,4,6,7,8], "showIndices": false, "highlightIndicesGreen": [0,1,2,3,4,5], "specialIndices": [] }

Array is fully sorted.

⬅ Previous TopicHeaps Technique in DSA

Next Topic ⮕Selection Sort