Counting Sort Algorithm

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

Counting sort is a sorting algorithm that sorts the elements of an array by counting the number of occurrences of each unique element in the array. The count is stored in an auxiliary array and the sorting is done by mapping the count as an index of the auxiliary array.


How Counting Sort Works?

  1. Find out the maximum element (let it be max) from the given array.
    Counting Sort steps
  2. Initialize an array of length max+1 with all elements 0. This array is used for storing the count of the elements in the array.
    Counting Sort Step

  3. Store the count of each element at their respective index in count array

    For example: If the count of element “4” occurs 2 times then 2 is stored in the 4th position in the count array. If element “5” is not present in the array, then 0 is stored in 5th position.
    Counting Sort Step
  4. Store cumulative sum of the elements of the count array.

    It helps in placing the elements into the correct index.

    If there are x elements less than y, its position should be at x-1.

    For example: In the array below, the count of 4 is 6. It denotes that there are 5 elements smaller than 4. Thus, the position of 4 in the sorted array is 5th.Counting Sort Step
  5. Find the index of each element of the original array in count array. This gives the cumulative count. Place the element at the index calculated.
    Counting Sort Steps
  6. After placing each element at its correct position, decrease the its count by one.

Counting Sort Algorithm

countingSort(array, size)
  max <- find largest element in array
  initialize count array with all zeros
  for j <- 0 to size
    find the total count of each unique element 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

# Counting sort in Python programming


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

    for i in range(0, size):
        count[array[i]] += 1

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

    i = size - 1
    while i >= 0:
        output[count[array[i]] - 1] = array[i]
        count[array[i]] -= 1
        i -= 1

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


data = [4, 2, 2, 8, 3, 3, 1]
countingSort(data)
print("Sorted Array in Ascending Order: ")
print(data)
// Counting sort in Java programming

import java.util.Arrays;

class CountingSort {
  void countSort(int array[], int size) {
    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]]++;
    }
    for (int i = 1; i <= max; i++) {
      count[i] += count[i - 1];
    }
    for (int i = size - 1; i >= 0; i--) {
      output[count[array[i]] - 1] = array[i];
      count[array[i]]--;
    }
    for (int i = 0; i < size; i++) {
      array[i] = output[i];
    }
  }

  public static void main(String args[]) {
    int[] data = { 4, 2, 2, 8, 3, 3, 1 };
    int size = data.length;
    CountingSort cs = new CountingSort();
    cs.countSort(data, size);
    System.out.println("Sorted Array in Ascending Order: ");
    System.out.println(Arrays.toString(data));
  }
}
// Counting sort in C programming

#include <stdio.h>

void countingSort(int array[], int size)
{
  int output[10];

  int max = array[0];
  for (int i = 1; i < size; i++)
  {
    if (array[i] > max)
      max = array[i];
  }
  // The size of count must be at least the (max+1) but
  // we cannot assign declare it as int count(max+1) in C as
  // it does not support dynamic memory allocation.
  // So, its size is provided statically.
  int count[10];
  for (int i = 0; i <= max; ++i)
  {
    count[i] = 0;
  }
  for (int i = 0; i < size; i++)
  {
    count[array[i]]++;
  }
  for (int i = 1; i <= max; i++)
  {
    count[i] += count[i - 1];
  }
  for (int i = size - 1; i >= 0; i--)
  {
    output[count[array[i]] - 1] = array[i];
    count[array[i]]--;
  }
  for (int i = 0; i < size; i++)
  {
    array[i] = output[i];
  }
}
void printArray(int array[], int size)
{
  for (int i = 0; i < size; ++i)
  {
    printf("%d  ", array[i]);
  }
  printf("\n");
}
int main()
{
  int array[] = {4, 2, 2, 8, 3, 3, 1};
  int n = sizeof(array) / sizeof(array[0]);
  countingSort(array, n);
  printArray(array, n);
}
// Counting sort in C++ programming

#include <iostream>
using namespace std;

void countSort(int array[], int size)
{
  // The size of count must be at least the (max+1) but
  // we cannot assign declare it as int count(max+1) in C++ as
  // it does not support dynamic memory allocation.
  // So, its size is provided statically.
  int output[10];
  int count[10];
  int max = array[0];
  for (int i = 1; i < size; i++)
  {
    if (array[i] > max)
      max = array[i];
  }

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

  for (int i = 0; i < size; i++)
  {
    count[array[i]]++;
  }
  for (int i = 1; i <= max; i++)
  {
    count[i] += count[i - 1];
  }
  for (int i = size - 1; i >= 0; i--)
  {
    output[count[array[i]] - 1] = array[i];
    count[array[i]]--;
  }
  for (int i = 0; i < size; i++)
  {
    array[i] = output[i];
  }
}
void printArray(int array[], int size)
{
  for (int i = 0; i < size; i++)
    cout << array[i] << " ";
  cout << endl;
}
int main()
{
  int array[] = {4, 2, 2, 8, 3, 3, 1};
  int n = sizeof(array) / sizeof(array[0]);
  countSort(array, n);
  printArray(array, n);
}

Complexity

Time Complexities:
There are mainly four main loops. (Finding the greatest value can be done outside the function.)

for-loop time of counting
1st O(max)
2nd O(size)
3rd O(max)
4th O(size)

Overall complexity = O(max)+O(size)+O(max)+O(size) = O(max+size)

  • Worst Case Complexity: O(n+k)
  • Best Case Complexity: O(n+k)
  • Average Case Complexity: O(n+k)

In all the above cases, the complexity is same because no matter how the elements are placed in the array, the algorithm goes through n+k times.

There is no comparison between any elements so, it is better than comparison based sorting techniques. But, it is bad if the integers are very large because the array of that size should be made.

Space Complexity:

The space complexity of Counting Sort is O(max). Larger the range of elements, larger is the space complexity.


Counting Sort Applications

Counting sort is used when:

  • the are smaller integers of multiple counts.
  • linear complexity is the need.