Suppose that we have an array of $n$ data records to sort and that the key of each record has the value $0$ or $1$. An algorithm for sorting such a set of records might possess some subset of the following three desirable characteristics:

1. The algorithm runs in $O(n)$ time.
2. The algorithm is stable.
3. The algorithm sorts in place, using no more than a constant amount of storage space in addition to the original array.

a. Give an algorithm that satisfies criteria 1 and 2 above.

b. Give an algorithm that satisfies criteria 1 and 3 above.

c. Give an algorithm that satisfies criteria 2 and 3 above.

d. Can you use any of your sorting algorithms from parts (a)–(c) as the sorting method used in line 2 of $\text{RADIX-SORT}$, so that $\text{RADIX-SORT}$ sorts $n$ records with $b$-bit keys in $O(bn)$ time? Explain how or why not.

e. Suppose that the $n$ records have keys in the range from $1$ to $k$. Show how to modify counting sort so that it sorts the records in place in $O(n + k)$ time. You may use $O(k)$ storage outside the input array. Is your algorithm stable? ($\textit{Hint:}$ How would you do it for $k = 3$?)

a. Counting-Sort.

b. Quicksort-Partition.

c. Insertion-Sort.

d. (a) Yes. (b) No. (c) No.

e.

Thanks @Gutdub for providing the solution in this issue.

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MODIFIED-COUNTING-SORT(A, k)
    let C[0..k] be a new array
    for i = 1 to k
        C[i] = 0
    for j = 1 to A.length
        C[A[j]] = C[A[j]] + 1
    for i = 2 to k
        C[i] = C[i] + C[i - 1]
    insert sentinel element NIL at the start of A
    B = C[0..k - 1]
    insert number 1 at the start of B
    // B now contains the "endpoints" for C
    for i = 2 to A.length
        while C[A[i]] != B[A[i]]
            key = A[i]
            exchange A[C[A[i]]] with A[i]
            while A[C[key]] == key // make sure that elements with the same keys will not be swapped
                C[key] = C[key] - 1
    remove the sentinel element
    return A

In place (storage space is $\Theta(k)$) but not stable.