Prerequisite: Counting Sort

QuickSort, MergeSort, HeapSort are comparison based sorting algorithms.
CountSort is not comparison based algorithm. It has the complexity of O(n+k), where k is the maximum element of the input array.
So, if k is O(n) ,CountSort becomes linear sorting, which is better than comparison based sorting algorithms that have O(nlogn) time complexity.
The idea is to extend the CountSort algorithm to get a better time complexity when k goes O(n2).
Here comes the idea of Radix Sort.

Algorithm:

For each digit i where i varies from the least significant digit to the most significant digit of a number
Sort input array using countsort algorithm according to ith digit.
We used count sort because it is a stable sort.

Example: Assume the input array is:

10, 21, 17, 34, 44, 11, 654, 123

Based on the algorithm, we will sort the input array according to the one’s digit (least significant digit).

0: 10

1: 21 11

2:

3: 123

4: 34 44 654

5:

6:

7: 17

8:

9:

So, the array becomes 10, 21, 11, 123, 24, 44, 654, 17
Now, we’ll sort according to the ten’s digit:

0:

1: 10 11 17

2: 21 123

3: 34

4: 44

5: 654

6:

7:

8:

9:

Now, the array becomes : 10, 11, 17, 21, 123, 34, 44, 654
Finally , we sort according to the hundred’s digit (most significant digit):

0: 010 011 017 021 034 044

1: 123

2:

3:

4:

5:

6: 654

7:

8:

9:

The array becomes : 10, 11, 17, 21, 34, 44, 123, 654 which is sorted. This is how our algorithm works.

An implementation in C:

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}```void countsort(int arr[],int n,int place){
int i,freq[range]={0}; // range for integers is 10 as digits range from 0-9
int output[n];

for(i=0;i<n;i++)
freq[(arr[i]/place)%range]++;

for(i=1;i<range;i++)
freq[i]+=freq[i-1];

for(i=n-1;i>=0;i--){
output[freq[(arr[i]/place)%range]-1]=arr[i];
freq[(arr[i]/place)%range]--;
}

for(i=0;i<n;i++)
arr[i]=output[i];
}

void radixsort(ll arr[],int n,int maxx){ // maxx is the maximum element in the array
int mul=1;
while(maxx){
countsort(arr,n,mul);
mul*=10;
maxx/=10;
}
}``````

An implementation in Python:

``````def counting_sort(arr, max_value, get_index):
counts =  * max_value

# Counting - O(n)
for a in arr:s
counts[get_index(a)] += 1

# Accumulating - O(k)
for i, c in enumerate(counts):
if i == 0:
continue
else:
counts[i] += counts[i-1]

# Calculating start index - O(k)
for i, c in enumerate(counts[:-1]):
if i == 0:
counts[i] = 0
counts[i+1] = c

ret = [None] * len(arr)
# Sorting - O(n)
for a in arr:
index = counts[get_index(a)]
ret[index] = a
counts[get_index(a)] += 1

return ret``````