Algorithm 1 Modified Counting Sort with Constant Extra Space

Input: a sequence $A$ of $n$ elements ranging from $0$ to $k$

Output: a sorted sequence $A$

1:let $B[0..k]$ be a new array initialized with all 0's

2:let $C[0..k]$ be a new array initialized with all 0's

3:for $i = 1$ to $n$ do

4: $B[A[i]] = B[A[i]] + 1$

5: $C[A[i]] = C[A[i]] + 1$

6:end for

7: // $B[i], C[i]$ now contains the number of elements equal to $i$

8:for $i = 1$ to $k$ do

9: $C[i] = C[i] + C[i-1]$

10:end for

11: // $C[i]$ now contains the number of elements less than or equal to $i$

12:for $i = 0$ to $k$ do

13:while $B[i] > 0$ do

14: $B[i] = B[i] - 1$

15: $A[C[i] - B[i]] = i$

16:end while

17:end for