Sorting is an essential task in computer science, and there are many sorting algorithms available, each with its advantages and disadvantages. One of the most efficient and straightforward sorting algorithms is the Radix Sort. In this article, we will explore how Radix Sort works, step-by-step, and provide Python code examples to implement it.

How Radix Sort Works

Radix Sort is a non-comparative, stable, and linear-time sorting algorithm that sorts data by grouping elements based on their significant digits or characters. It sorts the data by sorting each digit or character of the element individually, from the least significant digit to the most significant digit.

Let's take an example of sorting the following list of integers using Radix Sort:

**[170, 45, 75, 90, 802, 24, 2, 66]**

Determine the maximum number of digits in the list and pad the elements with zeroes if necessary. In our example, the maximum number of digits is three, so we will pad the elements with zeroes to make them all three digits long:

`[170, 045, 075, 090, 802, 024, 002, 066]`

Sort the list by the least significant digit (i.e., the rightmost digit). Create ten buckets (0-9) and place each element in the bucket corresponding to its digit. For example, 170, 090, and 802 have a 0 in the rightmost digit, so they will be placed in bucket 0. The list after this step will look like this:

`[802, 002, 024, 045, 075, 066, 170, 090]`

Concatenate the buckets in order, starting with bucket 0 and ending with bucket 9. The list after this step will look like this:

`[802, 002, 024, 045, 075, 066, 170, 090]`

Repeat steps 2 and 3 for the next significant digit (i.e., the second digit from the right), and continue until all the digits have been sorted. The final sorted list will be:

`[002, 024, 045, 066, 075, 090, 170, 802]`

Python Code for Radix Sort

Now that we understand how Radix Sort works let's implement it in Python. Here is the Python code for Radix Sort:

```
def radix_sort(nums):
# Determine the maximum number of digits
max_digit = len(str(max(nums)))
# Pad the elements with zeroes if necessary
nums = [str(num).zfill(max_digit) for num in nums]
# Sort the list by each digit
for i in range(max_digit - 1, -1, -1):
buckets = [[] for _ in range(10)]
for num in nums:
buckets[int(num[i])].append(num)
nums = [num for bucket in buckets for num in bucket]
# Convert the elements back to integers
nums = [int(num) for num in nums]
return nums
```

The code takes a list of integers as input and returns the sorted list using Radix Sort. It starts by determining the maximum number of digits in the list and pads the elements with zeroes if necessary. It then sorts the list by each digit, starting with the most significant digit, using the same steps we discussed earlier.

In addition to its simplicity and efficiency, one of the significant advantages of Radix Sort is its time complexity. The time complexity of Radix Sort is O(nk), where n is the number of elements to be sorted, and k is the maximum number of digits or characters in an element. This makes Radix Sort an efficient sorting algorithm for large datasets.

To see why the time complexity of Radix Sort is O(nk), let's consider the following:

In step 1, we need to determine the maximum number of digits or characters in the elements, which takes O(n) time.

In step 2, we need to iterate through each digit or character in each element, which takes O(kn) time.

In step 3, we need to concatenate the buckets, which takes O(n) time.

We repeat steps 2 and 3 for each digit or character in the elements, which takes O(kn) time.

Therefore, the total time complexity of Radix Sort is O(nk).

In conclusion, Radix Sort is a simple, efficient, and stable sorting algorithm that sorts data by grouping elements based on their significant digits or characters. With its time complexity of O(nk), it is an excellent choice for sorting large datasets. The Python code we provided in this article can be easily adapted to sort different types of data, including strings and tuples. By understanding how Radix Sort works and its time complexity, you can apply it to real-world scenarios and improve the performance of your applications.

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