Final answer:
Counting Sort involves an auxiliary array (B) and a counting array (C), with the range of A determining the size of C (k + 1 elements). The occurrences of each number in A are counted, accumulated, and used to determine the sorted positions. The result is a sorted array B = {1, 1, 3, 4, 4}.
Step-by-step explanation:
The operation of Counting Sort on the array A = {4, 4, 1, 3, 1} involves creating an auxiliary array B and a counting array C. First, we determine the range of values in A, which is from 1 to 4, so k is 4. Counting starts from 1 in this example and the arrays are zero-indexed.
Initialize C to have k + 1 = 5 elements to account for the value ranging from 1 to 4: C = {0, 0, 0, 0, 0}.
Step 1: Count the occurrences of each value in A and store them in C.
C = {0, 2, 0, 1, 2} (2 occurrences of 1, 1 occurrence of 3, and 2 occurrences of 4).
Step 2: Accumulate the counts in C to get the positions.
C = {0, 2, 2, 3, 5}.
Step 3: Place the elements of A into the sorted positions in B.
As we go through A, for each element A[i], we place it into B at the index specified by C[A[i]] - 1, and then decrease C[A[i]] by 1.
B = {1, 1, 3, 4, 4} after sorting.