Recursive function to list every element from cell arrays in Matlab

Question

I am trying to get all possible combinations of elements from an array of vectors. For example, say, I have a cell array

C = {[1 2 3 4 5], [6:13], [14 15]}

then the output should be something like

out = {[1 6 14], [1 6 15], [1 7 14], [1 7 15],....,[5 13 15]}

I tried to use a recursive function to implement this, but the code below doesn't seem to work. How can I obtain the list of all these combinations?

function [out,i] = permuteTest2(a,i,N,l,out)
    if nargin == 0
        a={[1 2],[4 5],[7 8]};
        N = length(a);
        i = 1;
        out = [];
    end
    if i == N
        out = [out, a{i}(l)];
        return;
    else
        for k=i:N
            L = length(a{k});
            for l=1:L
                out =[out a{k}(l)];
                [out,i] =permuteTest2(a, i+1, N,l,out);
            end
        end
    end

Please also edit the post on why you want to do this. As I said: for small problems this is trivial, whereas for slightly larger problems this quickly becomes impossible. For only 30 cells with 2 numbers each you'd need 2 full weeks of processing of processing, when a single permutation takes only a millisecond! Thus, why do you need all permutations? This smells of an optimisation problem, for which (much much) better solutions than brute forcing a grid search exist, exactly because calculating all possibilities takes infeasibly long. — Adriaan
– Adriaan, Commented Sep 7, 2022 at 9:21
You could select a random permutation as follows: for ii = 1:length(C); out(ii) = C{ii}(randi(numel(C{ii})));end, i.e. for each element in C, pick a random index and store the number on that index. — Adriaan
– Adriaan, Commented Sep 7, 2022 at 9:33
Your title asks for "every element" but your question asks for "a possible combination", but then you've shown an example of all combinations... which is it? Do you just want a random element from your example out, or do you want the complete set of permutations? — Wolfie
– Wolfie, Commented Sep 7, 2022 at 9:50

Cris Luengo · Accepted Answer · 2022-09-07 19:35:59Z

Say you have a cell array C with N vectors in it. To get all combinations of one element from each vector, you need to combine each element from the first vector with all combinations of the remaining vectors:

function combs = all_combinations(C)
   a = C{1};
   b = all_combinations(C(2:end));
   % ... get all the combinations of a and b here

If C has only one element, then all combinations is just that element (let's convert it to a cell array with one number in each cell, to match the expected output format):

function combs = all_combinations(C)
   a = C{1};
   if numel(C) == 1
      combs = num2cell(a);
      return;
   end
   b = all_combinations(C(2:end));
   % ... get all the combinations of a and b here

This takes care of the recursive part. Now all we need to do is find all combinations of the two sets a and b:

function combs = all_combinations(C)
   a = C{1};
   if numel(C) == 1
      combs = num2cell(a);
      return;
   end
   b = all_combinations(C(2:end));
   combs = cell(numel(a), numel(b));
   for ii = 1:numel(a)
      for jj = 1:numel(b)
         combs{ii, jj} = [a(ii), b{jj}];
      end
   end
   combs = combs(:);

To understand the answer above, it is important to track which elements are cell arrays and which are not, and how we index cell arrays ({} to extract an element from the array, () to index parts of the cell array, creating a new cell array).

I also used numel everywhere instead of length. numel is more efficient, and allows the code above to work equally well if instead of vectors you input arrays with more than one dimension.

A more robust function would test the input C to be a cell array with at least one element, and it would test each of the elements of this cell array to be a numeric array (or vector).

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