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Here's the problem. I've written a program that reads strings from stdin, and returns a random subset of those strings. The only other argument provided to the program is the length of the subset, $k$. The subset must contain exactly $k$ strings selected uniformly at random from the entire input set.

It's easy to do this if every single string is stored in memory. (Memory proportional to N). The question is how to only store at most $k$ strings, and still ensure that the output is perfectly random.

I've tried to work it out with the following base case. Say $k = 1$.

> Subset 1
A
B
C

A will always be added, since the queue contains less than k items. If I only operate on what I know, which is the current number of items in the queue, the required length $k$, and the encountered strings $n$. So I've tried doing this: ($k$ - items in queue)/$n$.

Using that, the probability of A being replaced by B is $1/2$. Then the probability of B being replaced by C will be $1/3$. The problem there is that there's no way of reducing the probability of A being replaced before I've seen all of the strings.

I'm sure this question must have a clever answer. The problem of writing this Subset program to use memory $<= k$ is a bonus question on an Algs course I'm taking, and I really hope they wouldn't ask something unanswerable.

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I seem to remember that this is not possible. To clarify: do you want $k$ i.i.d. uniform draws from the whole set, or one $k$-subset uniformly drawn from all $k$-subsets? –  Raphael Jul 1 at 6:29
    
@Raphael I'm going to say the former, but I can't see how the latter differs. Every single element in the whole set should have an equal probability of being selected for the subset. –  user2845306 Jul 1 at 19:38
1  
Consider the set $\{a,b,c,d\}$, $k=2$ and a probability distribution with $p_{a,b} = p_{b,c} = p_{c,d} = p_{a,d} = 1/4$. Note that every element has the same odds of being drawn -- $1/2$ -- but there are pairs with probability zero, e.g. $(b,d)$, that is we don't have a uniform distribution over all $2$-subsets. See my answer here (the wrong part) for a similar fallacy. –  Raphael Jul 1 at 21:40

2 Answers 2

up vote 9 down vote accepted

Use reservoir sampling. This is a good description in Wikipedia, or in Knuth.

Let's start with the simple case, where $k=1$. You always have one string in memory. When you read the first string, you store it in memory. Each time you read a new string, you replace it with the one in memory with probability $1/i$, if this is the $i$th string you've read so far. At the end, output whatever is stored in memory. The end result is that each string in the input is equally likely to be output. See also Choosing an element from a set satisfying a predicate uniformly at random in $O(1)$ space for description of this approach (thank you, Juho!).

This extends to arbitrary $k$. See Wikipedia's description for details.

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This question is relevant too. –  Juho Jul 1 at 7:55
1  
@Juho, thank you! Good find. I've updated my answer to link to that question. –  D.W. Jul 1 at 16:35

This problem is covered in The Art of Computer Programming. I can't recall exactly where, but the algorithm is pretty easy to understand when you know the trick.

Let $l$ be the number of lines read so far. At each stage, you read the next line from stdin. Choose $r$ to be a random integer uniformly chosen from the range $[1,l+1]$. If $r \le k$, then discard line number $r$ from the collection that you've kept, and replace it with the line you just read. Otherwise, drop the line.

The pseudocode looks something like this:

for i := 1 to k
    read a line from stdin into L[i]
end for

l := k
while there are more lines left
    read a line from stdin into x
    r := random(1, l+1)
    if r <= k then
        L[r] := x
    end if
    l := l + 1
end while

Of course, in a robust implementation, you should deal with the case that there are fewer lines than $k$, and so on.

Proof of correctness is left as an exercise.

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