Radix Sort Algorithm

In this tutorial, you will learn how radix sort works. Also, you will find working examples of radix sort in C, C++, Java and Python.

Radix sort is a sorting technique that sorts the elements by first grouping the individual digits of same place value. Then, sort the elements according to their increasing/decreasing order.

Suppose, we have an array of 8 elements. First, we will sort elements based on the value of the unit place. Then, we will sort elements based on the value of the tenth place. This process goes on until the last significant place.

Let the initial array be [121, 432, 564, 23, 1, 45, 788]. It is sorted according to radix sort as shown in the figure below.
Radix Sort Working

Please go through the counting sort before reading this article because counting sort is used as an intermediate sort in radix sort.


How Radix Sort Works?

  1. Find the largest element in the array, i.e. max. Let X be the number of digits in max. X is calculated because we have to go through all the significant places of all elements.

    In this array [121, 432, 564, 23, 1, 45, 788], we have the largest number 788. It has 3 digits. Therefore, the loop should go up to hundreds place (3 times).
  2. Now, go through each significant place one by one.

    Use any stable sorting technique to sort the digits at each significant place. We have used counting sort for this.

    Sort the elements based on the unit place digits (X=0).
    Radix Sort working with Counting Sort as intermediate step
  3. Now, sort the elements based on digits at tens place.
    Radix Sort Step
  4. Finally, sort the elements based on the digits at hundreds place.
    Selection Sort Step

Radix Sort Algorithm

radixSort(array)
  d <- maximum number of digits in the largest element
  create d buckets of size 0-9
  for i <- 0 to d
    sort the elements according to ith place digits using countingSort

countingSort(array, d)
  max <- find largest element among dth place elements
  initialize count array with all zeros
  for j <- 0 to size
    find the total count of each unique digit in dth place of elements and
    store the count at jth index in count array
  for i <- 1 to max
    find the cumulative sum and store it in count array itself
  for j <- size down to 1
    restore the elements to array
    decrease count of each element restored by 1

Python, Java and C/C++ Examples

# Radix sort in Python


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

    for i in range(0, size):
        index = array[i] // place
        count[index % 10] += 1

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

    i = size - 1
    while i >= 0:
        index = array[i] // place
        output[count[index % 10] - 1] = array[i]
        count[index % 10] -= 1
        i -= 1

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


def radixSort(array):
    max_element = max(array)
    place = 1
    while max_element // place > 0:
        countingSort(array, place)
        place *= 10


data = [121, 432, 564, 23, 1, 45, 788]
radixSort(data)
print(data)
// Radix Sort in Java Programming

import java.util.Arrays;

class RadixSort {
  void countingSort(int array[], int size, int place) {
    int[] output = new int[size + 1];
    int max = array[0];
    for (int i = 1; i < size; i++) {
      if (array[i] > max)
        max = array[i];
    }
    int[] count = new int[max + 1];

    for (int i = 0; i < max; ++i)
      count[i] = 0;

    for (int i = 0; i < size; i++)
      count[(array[i] / place) % 10]++;

    for (int i = 1; i < 10; i++)
      count[i] += count[i - 1];

    for (int i = size - 1; i >= 0; i--) {
      output[count[(array[i] / place) % 10] - 1] = array[i];
      count[(array[i] / place) % 10]--;
    }

    for (int i = 0; i < size; i++)
      array[i] = output[i];
  }

  int getMax(int array[], int n) {
    int max = array[0];
    for (int i = 1; i < n; i++)
      if (array[i] > max)
        max = array[i];
    return max;
  }

  void radixSort(int array[], int size) {
    int max = getMax(array, size);

    for (int place = 1; max / place > 0; place *= 10)
      countingSort(array, size, place);
  }

  public static void main(String args[]) {
    int[] data = { 121, 432, 564, 23, 1, 45, 788 };
    int size = data.length;
    RadixSort rs = new RadixSort();
    rs.radixSort(data, size);
    System.out.println("Sorted Array in Ascending Order: ");
    System.out.println(Arrays.toString(data));
  }
}
// Radix Sort in C Programming

#include <stdio.h>

int getMax(int array[], int n)
{
  int max = array[0];
  for (int i = 1; i < n; i++)
    if (array[i] > max)
      max = array[i];
  return max;
}
void countingSort(int array[], int size, int place)
{
  int output[size + 1];
  int max = (array[0] / place) % 10;

  for (int i = 1; i < size; i++)
  {
    if (((array[i] / place) % 10) > max)
      max = array[i];
  }
  int count[max + 1];

  for (int i = 0; i < max; ++i)
    count[i] = 0;

  for (int i = 0; i < size; i++)
    count[(array[i] / place) % 10]++;

  for (int i = 1; i < 10; i++)
    count[i] += count[i - 1];

  for (int i = size - 1; i >= 0; i--)
  {
    output[count[(array[i] / place) % 10] - 1] = array[i];
    count[(array[i] / place) % 10]--;
  }

  for (int i = 0; i < size; i++)
    array[i] = output[i];
}
void radixsort(int array[], int size)
{
  int max = getMax(array, size);

  for (int place = 1; max / place > 0; place *= 10)
    countingSort(array, size, place);
}
void printArray(int array[], int size)
{
  for (int i = 0; i < size; ++i)
  {
    printf("%d  ", array[i]);
  }
  printf("\n");
}
int main()
{
  int array[] = {121, 432, 564, 23, 1, 45, 788};
  int n = sizeof(array) / sizeof(array[0]);
  radixsort(array, n);
  printArray(array, n);
}
// Radix Sort in C++ Programming

#include <iostream>
using namespace std;

int getMax(int array[], int n)
{
  int max = array[0];
  for (int i = 1; i < n; i++)
    if (array[i] > max)
      max = array[i];
  return max;
}
void countingSort(int array[], int size, int place)
{
  const int max = 10;
  int output[size];
  int count[max];

  for (int i = 0; i < max; ++i)
    count[i] = 0;

  for (int i = 0; i < size; i++)
    count[(array[i] / place) % 10]++;

  for (int i = 1; i < max; i++)
    count[i] += count[i - 1];

  for (int i = size - 1; i >= 0; i--)
  {
    output[count[(array[i] / place) % 10] - 1] = array[i];
    count[(array[i] / place) % 10]--;
  }

  for (int i = 0; i < size; i++)
    array[i] = output[i];
}
void radixsort(int array[], int size)
{
  int max = getMax(array, size);

  for (int place = 1; max / place > 0; place *= 10)
    countingSort(array, size, place);
}
void printArray(int array[], int size)
{
  int i;
  for (i = 0; i < size; i++)
    cout << array[i] << " ";
  cout << endl;
}
int main()
{
  int array[] = {121, 432, 564, 23, 1, 45, 788};
  int n = sizeof(array) / sizeof(array[0]);
  radixsort(array, n);
  printArray(array, n);
}

Complexity

Since radix sort is a non-comparative algorithm, it has advantages over comparative sorting algorithms.

For the radix sort that uses counting sort as an intermediate stable sort, the time complexity is O(d(n+k)).

Here, d is the number cycle and O(n+k) is the time complexity of counting sort.

Thus, radix sort has linear time complexity which is better than O(nlog n) of comparative sorting algorithms.

If we take very large digit numbers or the number of other bases like 32-bit and 64-bit numbers then it can perform in linear time however the intermediate sort takes large space.

This makes radix sort space inefficient. This is the reason why this sort is not used in software libraries.


Radix Sort Applications

Radix sort is implemented in

  • DC3 algorithm (Kärkkäinen-Sanders-Burkhardt) while making suffix array.
  • places where there are numbers in large range.