Radix Sort – Data Structures and Algorithms Tutorials

Last Updated : 25 Nov, 2024

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.

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.

Radix Sort Algorithm

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. It’s important to understand that the default implementation of counting sort is unstable i.e. same keys can be in a different order than the input array. To solve this problem, We can iterate the input array in reverse order to build the output array. This strategy helps us to keep the same keys in the same order as they appear in the input array.

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 <iostream> 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[] = { 170, 45, 75, 90, 802, 24, 2, 66 };     int n = sizeof(arr) / sizeof(arr[0]);      // Function Call     radixsort(arr, n);     print(arr, n);     return 0; }

#include <stdio.h>  // A utility function to get the 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) {     int output[n]; // Output array     int count[10] = {0}; // Initialize count array as 0      // Store count of occurrences in count[]     for (int 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 (int i = 1; i < 10; i++)         count[i] += count[i - 1];      // Build the output array     for (int 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 (int i = 0; i < n; i++)         arr[i] = output[i]; }  // The main function to sort arr[] of size  // n using Radix Sort void radixSort(int arr[], int n) {        // Find the maximum number to know      // the number of digits     int m = getMax(arr, n);       // Do counting sort for every digit     // exp is 10^i where i is the current      // digit number     for (int exp = 1; m / exp > 0; exp *= 10)         countSort(arr, n, exp); }  // A utility function to print an array void printArray(int arr[], int n) {     for (int i = 0; i < n; i++)         printf("%d ", arr[i]);     printf("\n"); }  // Driver code int main() {     int arr[] = {170, 45, 75, 90, 802, 24, 2, 66};     int n = sizeof(arr) / sizeof(arr[0]);      // Function call     radixSort(arr, n);     printArray(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);     } }

Python

# 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# 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

// Radix sort JavaScript implementation  "use strict";  // A utility function to get maximum value in arr[] function getMax(arr) {   const length = arr.length;   let mx = arr[0];   for (let i = 1; i < length; 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. function countSort(arr, exp) {   const length = arr.length;   let output = Array(length); // output array   let count = Array(10).fill(0, 0);    // Store count of occurrences in count[]   for (let i = 0; i < length; i++) {     const digit = Math.floor(arr[i] / exp) % 10;     count[digit]++;   }    // Change count[i] so that count[i] now contains   // actual position of this digit in output[]   for (let i = 1; i < 10; i++) {     count[i] += count[i - 1];   }    // Build the output array   for (let i = length - 1; i >= 0; i--) {     const digit = Math.floor(arr[i] / exp) % 10;     output[count[digit] - 1] = arr[i];     count[digit]--;   }    return output; }  // The main function to that sorts arr[] using Radix Sort function radixSort(arr) {   // Find the maximum number to know number of digits   const maxNumber = getMax(arr);   // Create a shallow copy where the sorted values will be kept   let sortedArr = [...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 (let exp = 1; Math.floor(maxNumber / exp) > 0; exp *= 10) {     // Get the Count sort iteration     const sortedIteration = countSort(sortedArr, exp);     sortedArr = sortedIteration;   }    return sortedArr; }  /*Driver Code*/ const arr = [170, 45, 75, 90, 802, 24, 2, 66];  // Function Call const sortedArr = radixSort(arr);  console.log(sortedArr.join(" "));  // This code is contributed by beeduhboodee

PHP

<?php // PHP implementation of Radix Sort    // A function to do counting sort of arr[]  // according to the digit represented by exp.  function countSort(&$arr, $n, $exp)  {      $output = array_fill(0, $n, 0); // output array      $count = array_fill(0, 10, 0);       // Store count of occurrences in count[]      for ($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  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 ?>

Dart

// Radix sort Dart implementation  /// A utility function to get the maximum value of a `List<int>` [array] int getMax(List<int> array) {   int max = array[0];    for (final it in array) {     if (it > max) {       max = it;     }   }    return max; }  /// A function to do counting sort of `List<int>` [array] according to the /// digit represented by [exp]. List<int> countSort(List<int> array, int exp) {   final length = array.length;   final outputArr = List.filled(length, 0);   // A list where index represents the digit and value represents the count of   // occurrences   final digitsCount = List.filled(10, 0);    // Store count of occurrences in digitsCount[]   for (final item in array) {     final digit = item ~/ exp % 10;     digitsCount[digit]++;   }    // Change digitsCount[i] so that digitsCount[i] now contains actual position   // of this digit in outputArr[]   for (int i = 1; i < 10; i++) {     digitsCount[i] += digitsCount[i - 1];   }    // Build the output array   for (int i = length - 1; i >= 0; i--) {     final item = array[i];     final digit = item ~/ exp % 10;     outputArr[digitsCount[digit] - 1] = item;     digitsCount[digit]--;   }    return outputArr; }  /// The main function to that sorts a `List<int>` [array] using Radix sort List<int> radixSort(List<int> array) {   // Find the maximum number to know number of digits   final maxNumber = getMax(array);   // Shallow copy of the input array   final sortedArr = List.of(array);    // 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; maxNumber ~/ exp > 0; exp *= 10) {     final sortedIteration = countSort(sortedArr, exp);     sortedArr.clear();     sortedArr.addAll(sortedIteration);   }    return sortedArr; }  void main() {   const array = [170, 45, 75, 90, 802, 24, 2, 66];    final sortedArray = radixSort(array);    print(sortedArray); }  // This code is contributed by beeduhboodee

Output

2 24 45 66 75 90 170 802

Complexity Analysis of Radix Sort:

Time Complexity:

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:

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.