Radix sort is a sorting algorithm that sorts the elements by first grouping the individual digits of the 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.

Algorithm

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.

for example in the 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).

3. Now, sort the elements based on digits at tens place.

4. Finally, sort the elements based on the digits at hundreds place.

Code Implementation

// Radix Sort in Java Programming
import java.util.Arrays;

class RadixSort {

  // Using counting sort to sort the elements in the basis of significant places
  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;

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

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

    // Place the elements in sorted order
    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];
  }

  // Function to get the largest element from an array
  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;
  }

  // Main function to implement radix sort
  void radixSort(int array[], int size) {
    // Get maximum element
    int max = getMax(array, size);

    // Apply counting sort to sort elements based on place value.
    for (int place = 1; max / place > 0; place *= 10)
      countingSort(array, size, place);
  }

  // Driver code
  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));
  }
}

Complexity analysis

Time Complexity

Best Case: O(n+k)

Average Case: O(n+k)

Worst Case: O(n+k)

Space Complexity: O(max)

Stability: Yes

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.