Cartesian product of row-indices in Matlab - matlab

Question

I have a binary matrix A of dimension mxn with m>n in Matlab. I want to construct a matrix B of dimension cxn listing row wise each element of the Cartesian product of the row indices of the ones contained in A. To be more clear consider the following example.
Example:
%m=4;
%n=3;
A=[1 0 1;
0 0 1;
1 1 0;
0 0 1];
%column 1: "1" are at rows {1,3}
%column 2: "1" are at row {3}
%column 3: "1" are at rows {1,2,4}
%Hence, the Cartesian product {1,3}x{3}x{1,2,4} is
%{(1,3,1),(1,3,2),(1,3,4),(3,3,1),(3,3,2),(3,3,4)}
%I construct B by disposing row-wise each 3-tuple in the Cartesian product
%c=6
B=[1 3 1;
1 3 2;
1 3 4;
3 3 1;
3 3 2;
3 3 4];

Answer 1

You can get the cartesian product with the combvec command, for your example:
A=[1 0 1;...
0 0 1;...
1 1 0;...
0 0 1];
[x y]=find(A);
B=combvec(x(y==1).',x(y==2).',x(y==3).').';
% B =
% 1 3 1
% 3 3 1
% 1 3 2
% 3 3 2
% 1 3 4
% 3 3 4
You can expand this to an unknown number of columns by using the associative property of the product.
[x y]=find(A);
u_y=unique(y);
B=x(y==u_y(1)).';
for i=2:length(u_y)
B=combvec(B, x(y==u_y(i)).');
end
B=B.';

Answer 2

One solution (without toolbox):
A= [1 0 1;
0 0 1;
1 1 0;
0 0 1];
[ii,jj] = find(A)
kk = unique(jj);
for i = 1:length(kk)
v{i} = ii(jj==kk(i));
end
t=cell(1,length(kk));
[t{:}]= ndgrid(v{:});
product = []
for i = 1:length(kk)
product = [product,t{i}(:)];
end

Answer 3

You can use use accumarray to obtain vectors with the row indices of nonzero elements for each column. This works for an arbitrary number of columns:
[ii, jj] = find(A);
vectors = accumarray(jj, ii, [], #(x){sort(x.')});
Then apply this answer to efficiently compute the Cartesian product of those vectors:
n = numel(vectors);
B = cell(1,n);
[B{end:-1:1}] = ndgrid(vectors{end:-1:1});
B = cat(n+1, B{:});
B = reshape(B,[],n);
In your example, this gives
B =
1 3 1
1 3 2
1 3 4
3 3 1
3 3 2
3 3 4

Answer 4

In short, I would use find to generate the indices needed for the Cartesian product and then use ndgrid to perform the Cartesian product of these indices. The code to do so is:
clear
close all
clc
A = [1 0 1;
0 0 1;
1 1 0;
0 0 1];
[row,col] = find(A);
[~,ia,~] = unique(col);
n_cols = size(A,2);
indices = cell(n_cols,1);
for ii = 1:n_cols-1
indices{ii} = row(ia(ii):ia(ii+1)-1);
end
indices{end} = row(ia(end):end);
cp_temp = cell(n_cols,1);
[cp_temp{:}] = ndgrid(indices{:});
cp = NaN(numel(cp_temp{1}),n_cols);
for ii = 1:n_cols
cp(:,ii) = cp_temp{ii}(:);
end
cp = sortrows(cp);
cp