Cube sort is an efficient sorting algorithm that works on real-world data. It is based on the idea of partitioning the data set into “cubes” and sorting the elements within each cube. The algorithm works by scanning the given data set and grouping elements into cubes. It then sorts each cube separately and then merges the sorted cubes together. Cube sort is one of the most efficient sorting algorithms and is often used in applications such as database sorting.

In this article, we will discuss the cube sort algorithm in detail, including its time and space complexity. We will also provide Python code examples to illustrate how the algorithm works.

## How Cube Sort Works

The cube sort algorithm works by dividing the given data set into cubes. Each cube is a subset of the data set that contains elements that are similar to each other. The elements in each cube are then sorted using the appropriate sorting algorithm. The sorted cubes are then merged together to form the sorted data set.

The cube sort algorithm begins by scanning the given data set and dividing it into cubes. This is done by first finding the minimum and maximum values in the data set. The difference between the minimum and maximum values is then divided into a predetermined number of cubes. This predetermined number can be changed to adjust the speed and accuracy of the sorting algorithm.

Once the cubes are determined, the elements in each cube are sorted using an appropriate sorting algorithm. For example, if the elements in the cube are integers, then a quick sort might be used. If the elements in the cube are strings, then a merge sort might be used. Once each cube is sorted, the cubes are then merged together to form the sorted data set.

## Time Complexity of Cube Sort

The time complexity of cube sort depends on the size of the data set and the sorting algorithm used to sort the elements within each cube. The time complexity of cube sort is O(n log n) in the best case, O(n log n) in the average case, and O(n2) in the worst case.

The best case time complexity of cube sort is O(n log n). This occurs when the elements in each cube are already sorted, so no further sorting is required. The average case time complexity of cube sort is also O(n log n). This occurs when the elements in each cube are randomly distributed and require sorting. The worst case time complexity of cube sort is O(n2). This occurs when the elements in each cube are sorted in reverse order and require a lot of sorting.

## Space Complexity of Cube Sort

The space complexity of cube sort is O(n). This is because the algorithm requires an additional array to store the sorted cubes. The space complexity remains constant regardless of the size of the data set.

## Python Code Examples

The following Python code examples show how the cube sort algorithm works.

```
# Example 1: Cube Sort
# Input list of numbers
numbers = [6, 4, 8, 2, 5, 9, 7, 1, 3]
# Find the minimum and maximum values
min_value = min(numbers)
max_value = max(numbers)
# Calculate the number of cubes
num_cubes = int(max_value - min_value + 1)
# Create a list of cubes
cubes = [[] for _ in range(num_cubes)]
# Partition the list into cubes
for num in numbers:
cubes[num - min_value].append(num)
# Sort each cube
for cube in cubes:
cube.sort()
# Merge the sorted cubes
sorted_numbers = []
for cube in cubes:
sorted_numbers.extend(cube)
# Print the sorted list
print(sorted_numbers)
```

Here is another example:

```
# Example 2: Cube Sort using Quick Sort
# Input list of numbers
numbers = [6, 4, 8, 2, 5, 9, 7, 1, 3]
# Find the minimum and maximum values
min_value = min(numbers)
max_value = max(numbers)
# Calculate the number of cubes
num_cubes = int(max_value - min_value + 1)
# Create a list of cubes
cubes = [[] for _ in range(num_cubes)]
# Partition the list into cubes
for num in numbers:
cubes[num - min_value].append(num)
# Sort each cube using Quick Sort
for cube in cubes:
quick_sort(cube, 0, len(cube) - 1)
# Merge the sorted cubes
sorted_numbers = []
for cube in cubes:
sorted_numbers.extend(cube)
# Print the sorted list
print(sorted_numbers)
```

## Conclusion

In this article, we discussed cube sort, a sorting algorithm that is based on the idea of partitioning the data set into cubes and then sorting the elements within each cube. We also discussed the time and space complexity of cube sort, as well as provided Python code examples to illustrate how the algorithm works. Cube sort is an efficient sorting algorithm that is often used in applications such as database sorting.

## Exercises

#### Write a Python program using cube sort to sort a list of numbers in ascending order.

```
# Input list of numbers
numbers = [6, 4, 8, 2, 5, 9, 7, 1, 3]
# Find the minimum and maximum values
min_value = min(numbers)
max_value = max(numbers)
# Calculate the number of cubes
num_cubes = int(max_value - min_value + 1)
# Create a list of cubes
cubes = [[] for _ in range(num_cubes)]
# Partition the list into cubes
for num in numbers:
cubes[num - min_value].append(num)
# Sort each cube
for cube in cubes:
cube.sort()
# Merge the sorted cubes
sorted_numbers = []
for cube in cubes:
sorted_numbers.extend(cube)
# Print the sorted list
print(sorted_numbers)
```

#### Write a Python program using cube sort to sort a list of strings in alphabetical order.

```
# Input list of strings
strings = ["cat", "dog", "bird", "fish", "mouse"]
# Find the minimum and maximum values
min_value = ord(min(strings))
max_value = ord(max(strings))
# Calculate the number of cubes
num_cubes = int(max_value - min_value + 1)
# Create a list of cubes
cubes = [[] for _ in range(num_cubes)]
# Partition the list into cubes
for string in strings:
cubes[ord(string) - min_value].append(string)
# Sort each cube using Merge Sort
for cube in cubes:
merge_sort(cube, 0, len(cube) - 1)
# Merge the sorted cubes
sorted_strings = []
for cube in cubes:
sorted_strings.extend(cube)
# Print the sorted list
print(sorted_strings)
```

#### Modify the Python code from Example 1 to sort the numbers in descending order.

```
# Input list of numbers
numbers = [6, 4, 8, 2, 5, 9, 7, 1, 3]
# Find the minimum and maximum values
min_value = min(numbers)
max_value = max(numbers)
# Calculate the number of cubes
num_cubes = int(max_value - min_value + 1)
# Create a list of cubes
cubes = [[] for _ in range(num_cubes)]
# Partition the list into cubes
for num in numbers:
cubes[num - min_value].append(num)
# Sort each cube
for cube in cubes:
cube.sort(reverse=True)
# Merge the sorted cubes
sorted_numbers = []
for cube in cubes:
sorted_numbers.extend(cube)
# Print the sorted list
print(sorted_numbers)
```

#### Write a Python program to calculate the time complexity of cube sort in the worst case.

```
def cube_sort_time_complexity(n):
# Worst case time complexity is O(n^2)
return n**2
# Test cases
print(cube_sort_time_complexity(5)) # 25
print(cube_sort_time_complexity(10)) # 100
```

#### Write a Python program to calculate the space complexity of cube sort.

```
def cube_sort_space_complexity(n):
# Space complexity is O(n)
return n
# Test cases
print(cube_sort_space_complexity(5)) # 5
print(cube_sort_space_complexity(10)) # 10
```