How do I create all k-combinations with repetitions of a given set (also called k-multicombinations or multisubsets) using MATLAB?
This is similar to the cartesian product, but two rows that only differ by their sorting should be considered the same (e.g. the vectors [1,1,2]=~=[1,2,1]
are considered to be the same), so generating the cartesian product and then applying unique(sort(cartesianProduct,2),'rows')
should yield the same results.
Example:
The call nmultichoosek(1:n,k)
should generate the following matrix:
nmultichoosek(1:3,3)
ans =
1 1 1
1 1 2
1 1 3
1 2 2
1 2 3
1 3 3
2 2 2
2 2 3
2 3 3
3 3 3
We can use the bijection mentioned in the wikipedia article, which maps combinations without repetition of type n+k-1 choose k
to k
-multicombinations of size n
. We generate the combinations without repetition and map them using bsxfun(@minus, nchoosek(1:n+k-1,k), 0:k-1);
. This results in the following function:
function combs = nmultichoosek(values, k)
%// Return number of multisubsets or actual multisubsets.
if numel(values)==1
n = values;
combs = nchoosek(n+k-1,k);
else
n = numel(values);
combs = bsxfun(@minus, nchoosek(1:n+k-1,k), 0:k-1);
combs = reshape(values(combs),[],k);
end