all combinations of k elements out of n

Question

Can somebody provide me a link or pseudocode of a function for finding all combinations of k elements out of n? possibly in STL. I don't need to compute n choose k, I need to list all vectors of numbers of size k.

Thanks

Answer 1

In C++ given the following routine:

template <typename Iterator>
inline bool next_combination(const Iterator first, Iterator k, const Iterator last)
{
   /* Credits: Thomas Draper */
   if ((first == last) || (first == k) || (last == k))
      return false;
   Iterator itr1 = first;
   Iterator itr2 = last;
   ++itr1;
   if (last == itr1)
      return false;
   itr1 = last;
   --itr1;
   itr1 = k;
   --itr2;
   while (first != itr1)
   {
      if (*--itr1 < *itr2)
      {
         Iterator j = k;
         while (!(*itr1 < *j)) ++j;
         std::iter_swap(itr1,j);
         ++itr1;
         ++j;
         itr2 = k;
         std::rotate(itr1,j,last);
         while (last != j)
         {
            ++j;
            ++itr2;
         }
         std::rotate(k,itr2,last);
         return true;
      }
   }
   std::rotate(first,k,last);
   return false;
}

You can then proceed to do the following:

// 9-choose-3 
std::string s = "123456789";
std::size_t k = 3;
do
{
   std::cout << std::string(s.begin(),s.begin() + k) << std::endl;
}
while(next_combination(s.begin(),s.begin() + k,s.end()));

Or for a std::vector of int's:

// 5-choose-3 
std::size_t n = 5;
std::size_t k = 3;

std::vector<int> ints;
for (int i = 0; i < n; ints.push_back(i++));

do
{
   for (int i = 0; i < k; ++i)
   {
      std::cout << ints[i];
   }
   std::cout << "\n";
}
while(next_combination(ints.begin(),ints.begin() + k,ints.end()));

Answer 2

http://howardhinnant.github.io/combinations.html

Search for "for_each_combination". If you find something faster, please let me know. Unlike other algorithms I often see, this one doesn't require the element type to be LessThanComparable.

Answer 3

Create an auxiliary vector with n - k zeros followed by k ones. A zero means the element in the original container is not included, whereas one means the element is included.

Now use std::next_permutation on the auxiliary vector to get the next combination.

Answer 4

Here is a lazy example of pseudocode that can get the job done...

void nChooseK(array[n],k){
    recurse("",array[n],k);      
}

void recurse(initialString,array[n],k){
    if(k == 0){
        print initialString;
        return;
     }
    for(i=0;i<n;i++){
        tmpArray = array[0...i-1]+array[i+1...];//the array without the object to remove
        recurse(initialString + array[i], tmpArray,k-1)
    }        
}

Answer 5

You could use std::next_permutation but it is n! and not n choose k. You could filter them after you created them. But this solution is O(n!), not really perfect. Here is the trial and error solution:

int factorial(int value)
{
    int result = 1;

    for(int i = 1; i <= value; i++)
    {
        result *= i;
    }

    return result;
}

std::set<std::set<int>> binomial_coefficient(std::vector<int> input, int k)
{
    std::set<std::set<int>> solutions;

    for(unsigned int i = 0; i < factorial(input.size()); i++)
    {
        std::next_permutation(input.begin(), input.end());

        solutions.insert(std::set<int>(input.begin(), input.begin() + k));
    }

    return solutions;
}