Exercise 5.3.7

Suppose we want to create a random sample of the set $\{1, 2, 3, \ldots, n\}$, that is, an $m$-element subset $S$, where $0 \le m \le n$, such that each $m$-subset is equally likely to be created. One way would be to set $A[i] = i$ for $i = 1, 2, 3, \ldots, n$, call RANDOMIZE-IN-PLACE(A), and then take just the first $m$ array elements. This method would make $n$ calls to the RANDOM procedure. If $n$ is much larger than $m$, we can create a random sample with fewer calls to RANDOM. Show that the following recursive procedure returns a random $m$-subset $S$ of $\{1, 2, \ldots, n\}$, in which each $m$-subset is equally likely, while making only $m$ calls to RANDOM:

if m == 0
    return ∅
    S = RANDOM-SAMPLE(m-1, n-1)
    i = RANDOM(1,n)
    if i ∈ S
        S = S ∪ {n}
        S = S ∪ {i}
    return S

Each combination should have a $1/\binom{n}{m}$ chance of showing up. Let's establish an invariant for RANDOM-SAMPLE(m,n). We are going to go with:

RANDOM-SAMPLE(m,n) returns a uniformly distributed combination.

We shall do induction on $m$. It's trivially so when $m$ is 1 (or 0). Let's assume the invariant holds for $m-1$. Let's see what happens when we pass $m$.

The recursive call returns an $m-1$ sample with uniform probability. There are two options - either the new $m$-subset includes $n$ or not.

If that happens (probability: $m/n$), the probability for each combination including $n$ is:

$$ \frac{m}{n}\bigg(1/\binom{n-1}{m-1}\bigg) = 1/\binom{n}{m} $$

If it does not (probability: $(n-m)/n$), it includes one of the $(n-m)$ numbers not present. The chance for each is:

$$ \frac{n-m}{n}\bigg(1/\binom{n-1}{m}\bigg) = 1/\binom{n}{m} $$