Understanding Radix Sort, With Examples

Sorting data is a fundamental task in computer science and programming. When dealing with large data sets, choosing an efficient sorting algorithm becomes critical for performance. One often overlooked sorting method is the radix sort algorithm.

Radix sort is a non-comparative sorting algorithm that can provide major efficiency gains when sorting integer and string data types. Unlike comparison sorts, it avoids expensive pairwise element comparisons by recursively grouping and sorting elements based on individual digit or character positions.

In this comprehensive guide, we will demystify radix sort with beginner-friendly explanations, detailed code examples, and comparisons to other common sorting approaches. Let‘s dive in to better understand this powerful algorithm.

How Radix Sort Works

Conceptually, radix sort works by grouping numbers according to their digits‘ place values. It starts sorting from the least significant digit, continuing to recursively sort groups based on more significant digits, stopping once the most significant digit has been sorted.

For example, to radix sort the number sequence [490, 455, 234, 12, 67], the algorithm would:

  1. Check the 1s place digits and group/sort numbers into buckets:

    • 0: [490, 455]
    • 2: [12]
    • 4: [234]
    • 7: [67]
  2. Check the 10s place digits and group/sort numbers:

    • 0: [12]
    • 4: [490, 455, 234]
    • 6: [67]
  3. Check the 100s place and finish sorting numbers into order:

    • [12, 67, 234, 455, 490]

By only checking one digit place per pass rather than comparing every number, radix reduces the total number of comparisons needed, resulting in improved efficiency for sorting integer data.

Radix Sort Step-By-Step Example

To better demonstrate, let‘s walk through a detailed example sorting the number sequence:

[121, 232, 14, 46]

Initial sequence: [121, 232, 14, 46]

Iteration 1 (Ones Place):

  • Bucket 0: [46]
  • Bucket 1: [14, 121]
  • Bucket 2: [232]
  • After this iteration: [46, 14, 121, 232]

Iteration 2 (Tens Place):

  • Bucket 0: [46, 14]
  • Bucket 1: [121]
  • Bucket 2: [232]
  • After this iteration: [14, 46, 121, 232]

Iteration 3 (Hundreds Place):

  • Sequence already sorted: [14, 46, 121, 232]

And we are done! In just 3 iterations, examining one digit place per iteration, we have now fully sorted the entire sequence.

Radix Sort Time Complexity Analysis

The runtime efficiency of radix sort comes from only making a single pass examining one digit place per iteration, rather than comparing every number to each other per iteration.

Some key characteristics on time complexity:

  • Best case performance: O(kn) – where n is number of elements and k is number of iterations needed
  • Worst case performance: O(kn) – same as best case
  • Space complexity: O(n+k)

Unlike comparison sorts, radix sort runtime does not grow quadratically with input size. This makes it highly efficient for sorting integers or string data types as datasets scale to extremely large sizes.

Radix Sort In Action: Python Code Example

To make the algorithm more concrete, here is a Python implementation with a helper CountingSort() function:

def CountingSort(arr, place):
    size = len(arr)
    output = [0] * size
    count = [0] * 10

    # Calculate count of elements
    for i in range(0, size):
        index = arr[i] // place
        count[index % 10] += 1

    # Calculate cumulative count
    for i in range(1,10):
        count[i] += count[i - 1]

    # Place the elements in sorted order
    i = size - 1
    while i >= 0:
        index = arr[i] // place
        output[count[index % 10] - 1] = arr[i]
        count[index % 10] -= 1
        i -= 1

    for i in range(0, size):
        arr[i] = output[i]

def RadixSort(arr):
    # Get maximum element
    max_element = max(arr)

    # Apply counting sort to sort elements based on place value.
    place = 1
    while max_element // place > 0:
        CountingSort(arr, place)
        place *= 10

This demonstrates the logic and flow behind a typical radix sort implementation. It utilizes a CountingSort() helper to stably sort elements based on a specific digit place in each round.

When Should You Use Radix Sort?

Radix sort is best suited for:

  • Sorting integers and floating point numbers
  • Sorting string data when taking advantage of lexiconographic ordering
  • Data with many repeated elements
  • Large data sets where O(n^2) comparison sorts are prohibitively expensive

It should be avoided in cases where the integer keys are very sparsely distributed over the key space or do not start from 0. In those cases, the runtime can degrade to O(n^2) and other algorithms like quicksort may be better optimized.

Radix Sort vs Other Sorting Algorithms

Quicksort: Widely used comparison sort that offers excellent performance on average. However, suffers from worst case O(n^2).

Merge Sort: Stable sort with O(n log n) worst case. But constant factors make it slower than radix or quicksort in most cases.

Heapsort: Fast in-place algorithm with O(n log n) runtime, but not stable.

Radix tends to beat the performance of these approaches when sorting integers and strings, making it a great choice for numeric data and databases.


I hope this article helped explain the inner workings of the radix sort algorithm! By recursively grouping elements based on digit places, radix exploits the base-n structure of numeric data types to gain huge efficiency improvements compared to general comparison sorts.

Radix sort serves as a fast, stable foundation in environments like data pipelines, financial systems or scientific computing where integer performance is critical. For software engineers, understanding both its strengths and limitations provides another useful tool for the sorting toolbox.

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