Algorithm to select a single, random combination of values?

Question

Say I have y distinct values and I want to select x of them at random. What\'s an efficient algorithm for doing this? I could just call rand(

Answer 1

If you really only need to generate combinations - where the order of elements does not matter - you may use combinadics as they are implemented e.g. here by James McCaffrey.

Contrast this with k-permutations, where the order of elements does matter.

In the first case (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1) are considered the same - in the latter, they are considered distinct, though they contain the same elements.

In case you need combinations, you may really only need to generate one random number (albeit it can be a bit large) - that can be used directly to find the m th combination. Since this random number represents the index of a particular combination, it follows that your random number should be between 0 and C(n,k). Calculating combinadics might take some time as well.

It might just not worth the trouble - besides Jerry's and Federico's answer is certainly simpler than implementing combinadics. However if you really only need a combination and you are bugged about generating the exact number of random bits that are needed and none more... ;-)

While it is not clear whether you want combinations or k-permutations, here is a C# code for the latter (yes, we could generate only a complement if x > y/2, but then we would have been left with a combination that must be shuffled to get a real k-permutation):

static class TakeHelper
{
    public static IEnumerable<T> TakeRandom<T>(
        this IEnumerable<T> source, Random rng, int count)
    {
        T[] items = source.ToArray();

        count = count < items.Length ? count : items.Length;

        for (int i = items.Length - 1 ; count-- > 0; i--)
        {
            int p = rng.Next(i + 1);
            yield return items[p];
            items[p] = items[i];
        }
    }
}

class Program
{
    static void Main(string[] args)
    {
        Random rnd = new Random(Environment.TickCount);
        int[] numbers = new int[] { 1, 2, 3, 4, 5, 6, 7 };
        foreach (int number in numbers.TakeRandom(rnd, 3))
        {
            Console.WriteLine(number);
        }
    }
}

Another, more elaborate implementation that generates k-permutations, that I had lying around and I believe is in a way an improvement over existing algorithms if you only need to iterate over the results. While it also needs to generate x random numbers, it only uses O(min(y/2, x)) memory in the process:

    /// <summary>
    /// Generates unique random numbers
    /// <remarks>
    /// Worst case memory usage is O(min((emax-imin)/2, num))
    /// </remarks>
    /// </summary>
    /// <param name="random">Random source</param>
    /// <param name="imin">Inclusive lower bound</param>
    /// <param name="emax">Exclusive upper bound</param>
    /// <param name="num">Number of integers to generate</param>
    /// <returns>Sequence of unique random numbers</returns>
    public static IEnumerable<int> UniqueRandoms(
        Random random, int imin, int emax, int num)
    {
        int dictsize = num;
        long half = (emax - (long)imin + 1) / 2;
        if (half < dictsize)
            dictsize = (int)half;
        Dictionary<int, int> trans = new Dictionary<int, int>(dictsize);
        for (int i = 0; i < num; i++)
        {
            int current = imin + i;
            int r = random.Next(current, emax);
            int right;
            if (!trans.TryGetValue(r, out right))
            {
                right = r;
            }
            int left;
            if (trans.TryGetValue(current, out left))
            {
                trans.Remove(current);
            }
            else
            {
                left = current;
            }
            if (r > current)
            {
                trans[r] = left;
            }
            yield return right;
        }
    }

The general idea is to do a Fisher-Yates shuffle and memorize the transpositions in the permutation. It was not published anywhere nor has it received any peer-review whatsoever. I believe it is a curiosity rather than having some practical value. Nonetheless I am very open to criticism and would generally like to know if you find anything wrong with it - please consider this (and adding a comment before downvoting).

Answer 2

A little suggestion: if x >> y/2, it's probably better to select at random y - x elements, then choose the complementary set.

Answer 3

Here is a simple way to do it which is only inefficient if Y is much larger than X.

void randomly_select_subset(
    int X, int Y,
    const int * inputs, int X, int * outputs
) {
    int i, r;
    for( i = 0; i < X; ++i ) outputs[i] = inputs[i];
    for( i = X; i < Y; ++i ) {
        r = rand_inclusive( 0, i+1 );
        if( r < i ) outputs[r] = inputs[i];
    }
}

Basically, copy the first X of your distinct values to your output array, and then for each remaining value, randomly decide whether or not to include that value.

The random number is further used to choose an element of our (mutable) output array to replace.

Answer 4

Robert Floyd invented a sampling algorithm for just such situations. It's generally superior to shuffling then grabbing the first x elements since it doesn't require O(y) storage. As originally written it assumes values from 1..N, but it's trivial to produce 0..N and/or use non-contiguous values by simply treating the values it produces as subscripts into a vector/array/whatever.

In pseuocode, the algorithm runs like this (stealing from Jon Bentley's Programming Pearls column "A sample of Brilliance").

initialize set S to empty
for J := N-M + 1 to N do
    T := RandInt(1, J)
    if T is not in S then
        insert T in S
    else
        insert J in S

That last bit (inserting J if T is already in S) is the tricky part. The bottom line is that it assures the correct mathematical probability of inserting J so that it produces unbiased results.

It's O(x)¹ and O(1) with regard to y, O(x) storage.

Note that, in accordance with the combinations tag in the question, the algorithm only guarantees equal probability of each element occuring in the result, not of their relative order in it.

---

¹_{O(x²) in the worst case for the hash map involved which can be neglected since it's a virtually nonexistent pathological case where all the values have the same hash}

Answer 5

If, for example, you have 2^64 distinct values, you can use a symmetric key algorithm (with a 64 bits block) to quickly reshuffle all combinations. (for example Blowfish).

for(i=0; i<x; i++)
   e[i] = encrypt(key, i)

This is not random in the pure sense but can be useful for your purpose. If you want to work with arbitrary # of distinct values following cryptographic techniques you can but it's more complex.

Answer 6

The trick is to use a variation of shuffle or in other words a partial shuffle.

function random_pick( a, n ) 
{
  N = len(a);
  n = min(n, N);
  picked = array_fill(0, n, 0); backup = array_fill(0, n, 0);
  // partially shuffle the array, and generate unbiased selection simultaneously
  // this is a variation on fisher-yates-knuth shuffle
  for (i=0; i<n; i++) // O(n) times
  { 
    selected = rand( 0, --N ); // unbiased sampling N * N-1 * N-2 * .. * N-n+1
    value = a[ selected ];
    a[ selected ] = a[ N ];
    a[ N ] = value;
    backup[ i ] = selected;
    picked[ i ] = value;
  }
  // restore partially shuffled input array from backup
  // optional step, if needed it can be ignored
  for (i=n-1; i>=0; i--) // O(n) times
  { 
    selected = backup[ i ];
    value = a[ N ];
    a[ N ] = a[ selected ];
    a[ selected ] = value;
    N++;
  }
  return picked;
}

NOTE the algorithm is strictly O(n) in both time and space, produces unbiased selections (it is a partial unbiased shuffling) and non-destructive on the input array (as a partial shuffle would be) but this is optional

adapted from here

update

another approach using only a single call to PRNG (pseudo-random number generator) in [0,1] by IVAN STOJMENOVIC, "ON RANDOM AND ADAPTIVE PARALLEL GENERATION OF COMBINATORIAL OBJECTS" (section 3), of O(N) (worst-case) complexity