Five unique, random numbers from a subset

Question

I know similar questions come up a lot and there's probably no definitive answer, but I want to generate five unique random numbers from a subset of numbers that is potentially infinite (maybe 0-20, or 0-1,000,000).
The only catch is that I don't want to have to run while loops or fill an array.

My current method is to simply generate five random numbers from a subset minus the last five numbers. If any of the numbers match each other, then they go to their respective place at the end of the subset. So if the fourth number matches any other number, it will bet set to the 4th from the last number.

Does anyone have a method that is "random enough" and doesn't involve costly loops or arrays?

Please keep in mind this a curiosity, not some mission-critical problem. I would appreciate it if everyone didn't post "why are you having this problem?" answers. I am just looking for ideas.
Thanks a lot!

Answer 1

Your best option is a loop, as in:

$max = 20;
$numels = 5;
$vals = array();
while (count($vals) < $numels) {
    $cur = rand(0, $max);
    if (!in_array($cur, $vals))
        $vals[] = $cur;
}

For small ranges, you can use array_rand:

$max = 20;
$numels = 5;
$range = range(0, $max);
$vals = array_rand($range, $numels);

You could also generate a number between 0 and max, another between 0 and max-1, ... between 0 and max-4. Then you would sum x to the n-th generated number where x is the number calculated in this fashion:

• Take the number generated in the n-th iteration and assign it to x
• if it's larger or equal to that generated in the first iteration, increment it
• if this new number is larger or equal to that generated (and corrected) in the second iteration, increment it
• ...
• if this new number is larger or equal to that generated (and corrected) in the (n-1)-th iteration increment it

The mapping is like this:

1 2 3 4 5 6 7 8 9 (take 4)
1 2 3 4 5 6 7 8 9 (gives 4)

1 2 3 4 5 6 7 8 (take 5)
1 2 3 5 6 7 8 9 (gives 6)

1 2 3 4 5 6 7 (take 6)
1 2 3 5 7 8 9 (gives 8)

1 2 3 4 5 6 (take 5)
1 2 3 5 7 9 (gives 7)

example, last extraction:
x = 5
x >= 4? x == 6
x >= 6? x == 7
x >= 8? x == 7

Answer 2

One random number call is enough.

If you want to choose a subset of 5 unique numbers in range 1-n, then select a random number in 1 to (n choose r).

Keep a 1-1 mapping from 1 to (n choose r) to the set of possible 5 element subsets, and you are done. This mapping is standard and can be found on the web, for instance here: http://msdn.microsoft.com/en-us/library/aa289166%28VS.71%29.aspx

As an example:

Consider the problem of generating a subset of two numbers from five numbers:

The possible 2 element subset of {1,..., 5} are

1. {1,2}
2. {1,3}
3. {1,4}
4. {1,5}

5. {2,3}
6. {2,4}
7. {2,5}

8. {3,4}
9. {3,5}

10. {4,5}

Now 5 choose 2 is 10.

So we select a random number from 1 to 10. Say we got 8. Now we generate the 8th element in the sequence above: which gives {3,4}, so the two numbers you want are 3 and 4.

The msdn page I linked to, shows you a method to generate the set, given the number. i.e. given 8, it gives back the set {3,4}.

Answer 3

Since you are just looking for different ideas here's one:

Call out to Random.org to generate the set of random numbers you need.

Answer 4

The general form of this question is really interesting. Should one select from a pool of elements (and remove them from the pool) or should one loop "while hitting" an already taken element?

As far as I can tell, the python library implementation for random.sample chooses at runtime between the two methods depending on the proportion of the size of the input list and the number of elements to select.

A comment from the source code:

    # When the number of selections is small compared to the
    # population, then tracking selections is efficient, requiring
    # only a small set and an occasional reselection.  For
    # a larger number of selections, the pool tracking method is
    # preferred since the list takes less space than the
    # set and it doesn't suffer from frequent reselections.

---

In the specific instance that the OP mentions however (selecting 5 numbers), I think that looping "while hitting a taken number" is ok, unless the pseudo random generator is broken.

Answer 5

If you know the size N then keep each number with probability 5/N generate a random number between 0 and 1 and if it is less than 5/N keep the item. Stop when we have 5 items.

If we don't know N use resorvoir sampling.