# Radix Sort – Data Structures and Algorithms Tutorials

Radix Sort is a linear sorting algorithm that sorts elements by processing them digit by digit. It is an efficient sorting algorithm for integers or strings with fixed-size keys.Â

Rather than comparing elements directly, Radix Sort distributes the elements into buckets based on each digit’s value. By repeatedly sorting the elements by their significant digits, from the least significant to the most significant, Radix Sort achieves the final sorted order.

The key idea behind Radix Sort is to exploit the concept of place value. It assumes that sorting numbers digit by digit will eventually result in a fully sorted list. Radix Sort can be performed using different variations, such as Least Significant Digit (LSD) Radix Sort or Most Significant Digit (MSD) Radix Sort.

## How does Radix Sort Algorithm work?

To perform radix sort on the array [170, 45, 75, 90, 802, 24, 2, 66], we follow these steps:

How does Radix Sort Algorithm work | Step 1

Step 1: Find the largest element in the array, which is 802. It has three digits, so we will iterate three times, once for each significant place.

Step 2: Sort the elements based on the unit place digits (X=0). We use a stable sorting technique, such as counting sort, to sort the digits at each significant place.

Sorting based on the unit place:

• Perform counting sort on the array based on the unit place digits.
• The sorted array based on the unit place is [170, 90, 802, 2, 24, 45, 75, 66].

How does Radix Sort Algorithm work | Step 2

Step 3: Sort the elements based on the tens place digits.

Sorting based on the tens place:

• Perform counting sort on the array based on the tens place digits.
• The sorted array based on the tens place is [802, 2, 24, 45, 66, 170, 75, 90].

How does Radix Sort Algorithm work | Step 3

Step 4: Sort the elements based on the hundreds place digits.

Sorting based on the hundreds place:

• Perform counting sort on the array based on the hundreds place digits.
• The sorted array based on the hundreds place is [2, 24, 45, 66, 75, 90, 170, 802].

How does Radix Sort Algorithm work | Step 4

Step 5: The array is now sorted in ascending order.

The final sorted array using radix sort is [2, 24, 45, 66, 75, 90, 170, 802].

How does Radix Sort Algorithm work | Step 5

Below is the implementation for the above illustrations:

## C++

 `// C++ implementation of Radix Sort` `#include ``using` `namespace` `std;` `// A utility function to get maximum``// value in arr[]``int` `getMax(``int` `arr[], ``int` `n)``{``    ``int` `mx = arr[0];``    ``for` `(``int` `i = 1; i < n; i++)``        ``if` `(arr[i] > mx)``            ``mx = arr[i];``    ``return` `mx;``}` `// A function to do counting sort of arr[]``// according to the digit``// represented by exp.``void` `countSort(``int` `arr[], ``int` `n, ``int` `exp``)``{` `    ``// Output array``    ``int` `output[n];``    ``int` `i, count[10] = { 0 };` `    ``// Store count of occurrences``    ``// in count[]``    ``for` `(i = 0; i < n; i++)``        ``count[(arr[i] / ``exp``) % 10]++;` `    ``// Change count[i] so that count[i]``    ``// now contains actual position``    ``// of this digit in output[]``    ``for` `(i = 1; i < 10; i++)``        ``count[i] += count[i - 1];` `    ``// Build the output array``    ``for` `(i = n - 1; i >= 0; i--) {``        ``output[count[(arr[i] / ``exp``) % 10] - 1] = arr[i];``        ``count[(arr[i] / ``exp``) % 10]--;``    ``}` `    ``// Copy the output array to arr[],``    ``// so that arr[] now contains sorted``    ``// numbers according to current digit``    ``for` `(i = 0; i < n; i++)``        ``arr[i] = output[i];``}` `// The main function to that sorts arr[]``// of size n using Radix Sort``void` `radixsort(``int` `arr[], ``int` `n)``{` `    ``// Find the maximum number to``    ``// know number of digits``    ``int` `m = getMax(arr, n);` `    ``// Do counting sort for every digit.``    ``// Note that instead of passing digit``    ``// number, exp is passed. exp is 10^i``    ``// where i is current digit number``    ``for` `(``int` `exp` `= 1; m / ``exp` `> 0; ``exp` `*= 10)``        ``countSort(arr, n, ``exp``);``}` `// A utility function to print an array``void` `print(``int` `arr[], ``int` `n)``{``    ``for` `(``int` `i = 0; i < n; i++)``        ``cout << arr[i] << ``" "``;``}` `// Driver Code``int` `main()``{``    ``int` `arr[] = { 543, 986, 217, 765, 329 };``    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr[0]);` `    ``// Function Call``    ``radixsort(arr, n);``    ``print(arr, n);``    ``return` `0;``}`

## Java

 `// Radix sort Java implementation` `import` `java.io.*;``import` `java.util.*;` `class` `Radix {` `    ``// A utility function to get maximum value in arr[]``    ``static` `int` `getMax(``int` `arr[], ``int` `n)``    ``{``        ``int` `mx = arr[``0``];``        ``for` `(``int` `i = ``1``; i < n; i++)``            ``if` `(arr[i] > mx)``                ``mx = arr[i];``        ``return` `mx;``    ``}` `    ``// A function to do counting sort of arr[] according to``    ``// the digit represented by exp.``    ``static` `void` `countSort(``int` `arr[], ``int` `n, ``int` `exp)``    ``{``        ``int` `output[] = ``new` `int``[n]; ``// output array``        ``int` `i;``        ``int` `count[] = ``new` `int``[``10``];``        ``Arrays.fill(count, ``0``);` `        ``// Store count of occurrences in count[]``        ``for` `(i = ``0``; i < n; i++)``            ``count[(arr[i] / exp) % ``10``]++;` `        ``// Change count[i] so that count[i] now contains``        ``// actual position of this digit in output[]``        ``for` `(i = ``1``; i < ``10``; i++)``            ``count[i] += count[i - ``1``];` `        ``// Build the output array``        ``for` `(i = n - ``1``; i >= ``0``; i--) {``            ``output[count[(arr[i] / exp) % ``10``] - ``1``] = arr[i];``            ``count[(arr[i] / exp) % ``10``]--;``        ``}` `        ``// Copy the output array to arr[], so that arr[] now``        ``// contains sorted numbers according to current``        ``// digit``        ``for` `(i = ``0``; i < n; i++)``            ``arr[i] = output[i];``    ``}` `    ``// The main function to that sorts arr[] of``    ``// size n using Radix Sort``    ``static` `void` `radixsort(``int` `arr[], ``int` `n)``    ``{``        ``// Find the maximum number to know number of digits``        ``int` `m = getMax(arr, n);` `        ``// Do counting sort for every digit. Note that``        ``// instead of passing digit number, exp is passed.``        ``// exp is 10^i where i is current digit number``        ``for` `(``int` `exp = ``1``; m / exp > ``0``; exp *= ``10``)``            ``countSort(arr, n, exp);``    ``}` `    ``// A utility function to print an array``    ``static` `void` `print(``int` `arr[], ``int` `n)``    ``{``        ``for` `(``int` `i = ``0``; i < n; i++)``            ``System.out.print(arr[i] + ``" "``);``    ``}` `    ``// Main driver method``    ``public` `static` `void` `main(String[] args)``    ``{``        ``int` `arr[] = { ``170``, ``45``, ``75``, ``90``, ``802``, ``24``, ``2``, ``66` `};``        ``int` `n = arr.length;` `        ``// Function Call``        ``radixsort(arr, n);``        ``print(arr, n);``    ``}``}`

## Python3

 `# Python program for implementation of Radix Sort``# A function to do counting sort of arr[] according to``# the digit represented by exp.`  `def` `countingSort(arr, exp1):` `    ``n ``=` `len``(arr)` `    ``# The output array elements that will have sorted arr``    ``output ``=` `[``0``] ``*` `(n)` `    ``# initialize count array as 0``    ``count ``=` `[``0``] ``*` `(``10``)` `    ``# Store count of occurrences in count[]``    ``for` `i ``in` `range``(``0``, n):``        ``index ``=` `arr[i] ``/``/` `exp1``        ``count[index ``%` `10``] ``+``=` `1` `    ``# Change count[i] so that count[i] now contains actual``    ``# position of this digit in output array``    ``for` `i ``in` `range``(``1``, ``10``):``        ``count[i] ``+``=` `count[i ``-` `1``]` `    ``# Build the output array``    ``i ``=` `n ``-` `1``    ``while` `i >``=` `0``:``        ``index ``=` `arr[i] ``/``/` `exp1``        ``output[count[index ``%` `10``] ``-` `1``] ``=` `arr[i]``        ``count[index ``%` `10``] ``-``=` `1``        ``i ``-``=` `1` `    ``# Copying the output array to arr[],``    ``# so that arr now contains sorted numbers``    ``i ``=` `0``    ``for` `i ``in` `range``(``0``, ``len``(arr)):``        ``arr[i] ``=` `output[i]` `# Method to do Radix Sort`  `def` `radixSort(arr):` `    ``# Find the maximum number to know number of digits``    ``max1 ``=` `max``(arr)` `    ``# Do counting sort for every digit. Note that instead``    ``# of passing digit number, exp is passed. exp is 10^i``    ``# where i is current digit number``    ``exp ``=` `1``    ``while` `max1 ``/` `exp >``=` `1``:``        ``countingSort(arr, exp)``        ``exp ``*``=` `10`  `# Driver code``arr ``=` `[``170``, ``45``, ``75``, ``90``, ``802``, ``24``, ``2``, ``66``]` `# Function Call``radixSort(arr)` `for` `i ``in` `range``(``len``(arr)):``    ``print``(arr[i], end``=``" "``)` `# This code is contributed by Mohit Kumra``# Edited by Patrick Gallagher`

## C#

 `// C# implementation of Radix Sort``using` `System;` `class` `GFG {``    ``public` `static` `int` `getMax(``int``[] arr, ``int` `n)``    ``{``        ``int` `mx = arr[0];``        ``for` `(``int` `i = 1; i < n; i++)``            ``if` `(arr[i] > mx)``                ``mx = arr[i];``        ``return` `mx;``    ``}` `    ``// A function to do counting sort of arr[] according to``    ``// the digit represented by exp.``    ``public` `static` `void` `countSort(``int``[] arr, ``int` `n, ``int` `exp)``    ``{``        ``int``[] output = ``new` `int``[n]; ``// output array``        ``int` `i;``        ``int``[] count = ``new` `int``[10];` `        ``// initializing all elements of count to 0``        ``for` `(i = 0; i < 10; i++)``            ``count[i] = 0;` `        ``// Store count of occurrences in count[]``        ``for` `(i = 0; i < n; i++)``            ``count[(arr[i] / exp) % 10]++;` `        ``// Change count[i] so that count[i] now contains``        ``// actual``        ``//  position of this digit in output[]``        ``for` `(i = 1; i < 10; i++)``            ``count[i] += count[i - 1];` `        ``// Build the output array``        ``for` `(i = n - 1; i >= 0; i--) {``            ``output[count[(arr[i] / exp) % 10] - 1] = arr[i];``            ``count[(arr[i] / exp) % 10]--;``        ``}` `        ``// Copy the output array to arr[], so that arr[] now``        ``// contains sorted numbers according to current``        ``// digit``        ``for` `(i = 0; i < n; i++)``            ``arr[i] = output[i];``    ``}` `    ``// The main function to that sorts arr[] of size n using``    ``// Radix Sort``    ``public` `static` `void` `radixsort(``int``[] arr, ``int` `n)``    ``{``        ``// Find the maximum number to know number of digits``        ``int` `m = getMax(arr, n);` `        ``// Do counting sort for every digit. Note that``        ``// instead of passing digit number, exp is passed.``        ``// exp is 10^i where i is current digit number``        ``for` `(``int` `exp = 1; m / exp > 0; exp *= 10)``            ``countSort(arr, n, exp);``    ``}` `    ``// A utility function to print an array``    ``public` `static` `void` `print(``int``[] arr, ``int` `n)``    ``{``        ``for` `(``int` `i = 0; i < n; i++)``            ``Console.Write(arr[i] + ``" "``);``    ``}` `    ``// Driver Code``    ``public` `static` `void` `Main()``    ``{``        ``int``[] arr = { 170, 45, 75, 90, 802, 24, 2, 66 };``        ``int` `n = arr.Length;` `        ``// Function Call``        ``radixsort(arr, n);``        ``print(arr, n);``    ``}` `    ``// This code is contributed by DrRoot_``}`

## Javascript

 ``

## PHP

 `= 0; ``\$i``--) ``    ``{ ``        ``\$output``[``\$count``[ (``\$arr``[``\$i``] / ``                         ``\$exp``) % 10 ] - 1] = ``\$arr``[``\$i``]; ``        ``\$count``[ (``\$arr``[``\$i``] / ``\$exp``) % 10 ]--; ``    ``} ` `    ``// Copy the output array to arr[], so ``    ``// that arr[] now contains sorted numbers``    ``// according to current digit ``    ``for` `(``\$i` `= 0; ``\$i` `< ``\$n``; ``\$i``++) ``        ``\$arr``[``\$i``] = ``\$output``[``\$i``]; ``} ` `// The main function to that sorts arr[] ``// of size n using Radix Sort ``function` `radixsort(&``\$arr``, ``\$n``) ``{ ``    ` `    ``// Find the maximum number to know``    ``// number of digits ``    ``\$m` `= max(``\$arr``); ` `    ``// Do counting sort for every digit. Note ``    ``// that instead of passing digit number, ``    ``// exp is passed. exp is 10^i where i is ``    ``// current digit number ``    ``for` `(``\$exp` `= 1; ``\$m` `/ ``\$exp` `> 0; ``\$exp` `*= 10) ``        ``countSort(``\$arr``, ``\$n``, ``\$exp``); ``} ` `// A utility function to print an array ``function` `PrintArray(&``\$arr``,``\$n``) ``{ ``    ``for` `(``\$i` `= 0; ``\$i` `< ``\$n``; ``\$i``++) ``        ``echo` `\$arr``[``\$i``] . ``" "``; ``} ` `// Driver Code ``\$arr` `= ``array``(170, 45, 75, 90, 802, 24, 2, 66); ``\$n` `= ``count``(``\$arr``); ` `// Function Call``radixsort(``\$arr``, ``\$n``); ``PrintArray(``\$arr``, ``\$n``); ` `// This code is contributed by rathbhupendra``?>`

Output
```217 329 543 765 986
```

## Complexity Analysis of Radix Sort:

Time Complexity:Â

• Radix sort is a non-comparative integer sorting algorithm that sorts data with integer keys by grouping the keys by the individual digits which share the same significant position and value. It has a time complexity of O(d * (n + b)), where d is the number of digits, n is the number of elements, and b is the base of the number system being used.
• In practical implementations, radix sort is often faster than other comparison-based sorting algorithms, such as quicksort or merge sort, for large datasets, especially when the keys have many digits. However, its time complexity grows linearly with the number of digits, and so it is not as efficient for small datasets.

Auxiliary Space:Â

• Radix sort also has a space complexity of O(n + b), where n is the number of elements and b is the base of the number system. This space complexity comes from the need to create buckets for each digit value and to copy the elements back to the original array after each digit has been sorted.

Q1. Is Radix Sort preferable to Comparison based sorting algorithms like Quick-Sort?Â

If we have log2n bits for every digit, the running time of Radix appears to be better than Quick Sort for a wide range of input numbers. The constant factors hidden in asymptotic notation are higher for Radix Sort and Quick-Sort uses hardware caches more effectively. Also, Radix sort uses counting sort as a subroutine and counting sort takes extra space to sort numbers.

Q2. What if the elements are in the range from 1 to n2?

• The lower bound for the Comparison based sorting algorithm (Merge Sort, Heap Sort, Quick-Sort .. etc) is Î©(nLogn), i.e., they cannot do better than nLogn. Counting sort is a linear time sorting algorithm that sort in O(n+k) time when elements are in the range from 1 to k.
• We can’t use counting sort because counting sort will take O(n2) which is worse than comparison-based sorting algorithms. Can we sort such an array in linear time?Â
• Radix Sort is the answer. The idea of Radix Sort is to do digit-by-digit sorting starting from the least significant digit to the most significant digit. Radix sort uses counting sort as a subroutine to sort.