Counting Sort – Data Structures and Algorithms Tutorials

Last Updated : 09 Mar, 2024

What is Counting Sort?

Counting Sort is a non-comparison-based sorting algorithm that works well when there is limited range of input values. It is particularly efficient when the range of input values is small compared to the number of elements to be sorted. The basic idea behind Counting Sort is to count the frequency of each distinct element in the input array and use that information to place the elements in their correct sorted positions.

How does Counting Sort Algorithm work?

Step1 :

Find out the maximum element from the given array.

Finding maximum element in inputArray[]

Step 2:

Initialize a countArray[] of length max+1 with all elements as 0. This array will be used for storing the occurrences of the elements of the input array.

Initialize countArray[]

Step 3:

In the countArray[], store the count of each unique element of the input array at their respective indices.
For Example: The count of element 2 in the input array is 2. So, store 2 at index 2 in the countArray[]. Similarly, the count of element 5 in the input array is 1, hence store 1 at index 5 in the countArray[].

Maintain count of each element in countArray[]

Step 4:

Store the cumulative sum or prefix sum of the elements of the countArray[] by doing countArray[i] = countArray[i – 1] + countArray[i]. This will help in placing the elements of the input array at the correct index in the output array.

Store the cumulative sum in countArray[]

Step 5:

Iterate from end of the input array and because traversing input array from end preserves the order of equal elements, which eventually makes this sorting algorithm stable.

Update outputArray[ countArray[ inputArray[i] ] – 1] = inputArray[i].

Also, update countArray[ inputArray[i] ] = countArray[ inputArray[i] ]– -.

Step 6: For i = 6,

Update outputArray[ countArray[ inputArray[6] ] – 1] = inputArray[6]
Also, update countArray[ inputArray[6] ] = countArray[ inputArray[6] ]- –

Placing inputArray[6] at it correct position in outputArray[]

Step 7: For i = 5,

Update outputArray[ countArray[ inputArray[5] ] – 1] = inputArray[5]
Also, update countArray[ inputArray[5] ] = countArray[ inputArray[5] ]- –

Placing inputArray[5] at it correct position in outputArray[]

Step 8: For i = 4,

Update outputArray[ countArray[ inputArray[4] ] – 1] = inputArray[4]
Also, update countArray[ inputArray[4] ] = countArray[ inputArray[4] ]- –

Placing inputArray[4] at it correct position in outputArray[]

Step 9: For i = 3,

Update outputArray[ countArray[ inputArray[3] ] – 1] = inputArray[3]
Also, update countArray[ inputArray[3] ] = countArray[ inputArray[3] ]- –

Placing inputArray[3] at it correct position in outputArray[]

Step 10: For i = 2,

Update outputArray[ countArray[ inputArray[2] ] – 1] = inputArray[2]
Also, update countArray[ inputArray[2] ] = countArray[ inputArray[2] ]- –

Placing inputArray[2] at it correct position in outputArray[]

Step 11: For i = 1,

Update outputArray[ countArray[ inputArray[1] ] – 1] = inputArray[1]
Also, update countArray[ inputArray[1] ] = countArray[ inputArray[1] ]- –

Placing inputArray[1] at it correct position in outputArray[]

Step 12: For i = 0,

Update outputArray[ countArray[ inputArray[0] ] – 1] = inputArray[0]
Also, update countArray[ inputArray[0] ] = countArray[ inputArray[0] ]- –

Placing inputArray[0] at it correct position in outputArray[]

Counting Sort Algorithm:

Declare an auxiliary array countArray[] of size max(inputArray[])+1 and initialize it with 0.
Traverse array inputArray[] and map each element of inputArray[] as an index of countArray[] array, i.e., execute countArray[inputArray[i]]++ for 0 <= i < N.
Calculate the prefix sum at every index of array inputArray[].
Create an array outputArray[] of size N.
Traverse array inputArray[] from end and update outputArray[ countArray[ inputArray[i] ] – 1] = inputArray[i]. Also, update countArray[ inputArray[i] ] = countArray[ inputArray[i] ]- – .

Below is the implementation of the above algorithm:

Java

import java.util.Arrays;
 
public class CountSort {
    public static int[] countSort(int[] inputArray) {
        int N = inputArray.length;
        int M = 0;
 
        for (int i = 0; i < N; i++) {
            M = Math.max(M, inputArray[i]);
        }
 
        int[] countArray = new int[M + 1];
 
        for (int i = 0; i < N; i++) {
            countArray[inputArray[i]]++;
        }
 
        for (int i = 1; i <= M; i++) {
            countArray[i] += countArray[i - 1];
        }
 
        int[] outputArray = new int[N];
 
        for (int i = N - 1; i >= 0; i--) {
            outputArray[countArray[inputArray[i]] - 1] = inputArray[i];
            countArray[inputArray[i]]--;
        }
 
        return outputArray;
    }
 
    public static void main(String[] args) {
        int[] inputArray = {4, 3, 12, 1, 5, 5, 3, 9};
        int[] outputArray = countSort(inputArray);
 
        for (int i = 0; i < inputArray.length; i++) {
            System.out.print(outputArray[i] + " ");
        }
    }
}

C#

using System;
using System.Collections.Generic;
 
class GFG
{
    static List<int> CountSort(List<int> inputArray)
    {
        int N = inputArray.Count;
        // Finding the maximum element of the array inputArray[].
        int M = 0;
        for (int i = 0; i < N; i++)
            M = Math.Max(M, inputArray[i]);
        // Initializing countArray[] with 0
        List<int> countArray = new List<int>(new int[M + 1]);
        // Mapping each element of inputArray[] as an index
        // of countArray[] array
        for (int i = 0; i < N; i++)
            countArray[inputArray[i]]++;
        // Calculating prefix sum at every index
        // of array countArray[]
        for (int i = 1; i <= M; i++)
            countArray[i] += countArray[i - 1];
        // Creating outputArray[] from the countArray[] array
        List<int> outputArray = new List<int>(new int[N]);
        for (int i = N - 1; i >= 0; i--)
        {
            outputArray[countArray[inputArray[i]] - 1] = inputArray[i];
            countArray[inputArray[i]]--;
        }
        return outputArray;
    }
    // Driver code
    static void Main()
    {
        // Input array
        List<int> inputArray = new List<int> { 4, 3, 12, 1, 5, 5, 3, 9 };
        // Output array
        List<int> outputArray = CountSort(inputArray);
        for (int i = 0; i < inputArray.Count; i++)
            Console.Write(outputArray[i] + " ");
        Console.WriteLine();
    }
}

Javascript

function countSort(inputArray) {
    const N = inputArray.length;
 
    // Finding the maximum element of inputArray
    let M = 0;
    for (let i = 0; i < N; i++) {
        M = Math.max(M, inputArray[i]);
    }
 
    // Initializing countArray with 0
    const countArray = new Array(M + 1).fill(0);
 
    // Mapping each element of inputArray as an index of countArray
    for (let i = 0; i < N; i++) {
        countArray[inputArray[i]]++;
    }
 
    // Calculating prefix sum at every index of countArray
    for (let i = 1; i <= M; i++) {
        countArray[i] += countArray[i - 1];
    }
 
    // Creating outputArray from countArray
    const outputArray = new Array(N);
    for (let i = N - 1; i >= 0; i--) {
        outputArray[countArray[inputArray[i]] - 1] = inputArray[i];
        countArray[inputArray[i]]--;
    }
 
    return outputArray;
}
 
// Driver code
const inputArray = [4, 3, 12, 1, 5, 5, 3, 9];
 
// Sorting the input array
const outputArray = countSort(inputArray);
 
// Printing the sorted array
console.log(outputArray.join(' '));
//This code is contributed by Utkarsh

C++14

#include <bits/stdc++.h>
using namespace std;
 
vector<int> countSort(vector<int>& inputArray)
{
 
    int N = inputArray.size();
 
    // Finding the maximum element of array inputArray[].
    int M = 0;
 
    for (int i = 0; i < N; i++)
        M = max(M, inputArray[i]);
 
    // Initializing countArray[] with 0
    vector<int> countArray(M + 1, 0);
 
    // Mapping each element of inputArray[] as an index
    // of countArray[] array
 
    for (int i = 0; i < N; i++)
        countArray[inputArray[i]]++;
 
    // Calculating prefix sum at every index
    // of array countArray[]
    for (int i = 1; i <= M; i++)
        countArray[i] += countArray[i - 1];
 
    // Creating outputArray[] from countArray[] array
    vector<int> outputArray(N);
 
    for (int i = N - 1; i >= 0; i--)
 
    {
        outputArray[countArray[inputArray[i]] - 1]
            = inputArray[i];
 
        countArray[inputArray[i]]--;
    }
 
    return outputArray;
}
 
// Driver code
int main()
 
{
 
    // Input array
    vector<int> inputArray = { 4, 3, 12, 1, 5, 5, 3, 9 };
 
    // Output array
    vector<int> outputArray = countSort(inputArray);
 
    for (int i = 0; i < inputArray.size(); i++)
        cout << outputArray[i] << " ";
 
    return 0;
}

Python3

def count_sort(input_array):
    # Finding the maximum element of input_array.
    M = max(input_array)
 
    # Initializing count_array with 0
    count_array = [0] * (M + 1)
 
    # Mapping each element of input_array as an index of count_array
    for num in input_array:
        count_array[num] += 1
 
    # Calculating prefix sum at every index of count_array
    for i in range(1, M + 1):
        count_array[i] += count_array[i - 1]
 
    # Creating output_array from count_array
    output_array = [0] * len(input_array)
 
    for i in range(len(input_array) - 1, -1, -1):
        output_array[count_array[input_array[i]] - 1] = input_array[i]
        count_array[input_array[i]] -= 1
 
    return output_array
 
# Driver code
if __name__ == "__main__":
    # Input array
    input_array = [4, 3, 12, 1, 5, 5, 3, 9]
 
    # Output array
    output_array = count_sort(input_array)
 
    for num in output_array:
        print(num, end=" ")

Output

1 3 3 4 5 5 9 12

Complexity Analysis of Counting Sort:

Time Complexity: O(N+M), where N and M are the size of inputArray[] and countArray[] respectively.
- Worst-case: O(N+M).
- Average-case: O(N+M).
- Best-case: O(N+M).
Auxiliary Space: O(N+M), where N and M are the space taken by outputArray[] and countArray[] respectively.

Advantage of Counting Sort:

Counting sort generally performs faster than all comparison-based sorting algorithms, such as merge sort and quicksort, if the range of input is of the order of the number of input.
Counting sort is easy to code
Counting sort is a stable algorithm.

Disadvantage of Counting Sort:

Counting sort doesn’t work on decimal values.
Counting sort is inefficient if the range of values to be sorted is very large.
Counting sort is not an In-place sorting algorithm, It uses extra space for sorting the array elements.

Suggest improvement

Median and Mode using Counting Sort

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Problem based on Counting Sort