Finding all possible “lists” of possible pairs in Matlab - matlab

Question

I have been thinking about a problem for the last few days but as I am a beginner in MATLAB, I have no clue how to solve it. Here is the background. Suppose that you have a symmetric N×N matrix where each element is either 0 or 1, and N = (1,2,...,n).
For example:
A =
0 1 1 0
1 0 0 1
1 0 0 0
0 1 0 0
If A(i,j) == 1, then it is possible to form the pair (i,j) and if A(i,j)==0 then it is NOT possible to form the pair (i,j). For example, (1,2) is a possible pair, as A(1,2)==A(2,1)==1 but (3,4) is NOT a possible pair as A(3,4)==A(4,3)==0.
Here is the problem. Suppose that a member of the set N only can for a pair with at most one other distinct member of the set N (i.e., if 1 forms a pair with 2, then 1 cannot form a pair with 3). How can I find all possible “lists” of possible pairs? In the above example, one “list” would only consist of the pair (1,2). If this pair is formed, then it is not possible to form any other pairs. Another “list” would be: ((1,3),(2,4)). I have searched the forum and found that the latter “list” is the maximal matching that can be found, e.g., by using a bipartite graph approach. However, I am not necessarily only interested to find the maximal matching; I am interested in finding ALL possible “lists” of possible pairs.
Another example:
A =
0 1 1 1
1 0 0 1
1 0 0 0
1 1 0 0
In this example, there are three possible lists:
(1,2)
((1,3),(2,4))
(1,4)
I hope that you can understand my question, and I apologize if am unclear. I appreciate all help I can get. Many thanks!

Answer 1

This might be a fast approach.
Code
%// Given data, A
A =[ 0 1 1 1;
1 0 0 1;
1 0 0 0;
1 1 0 0];
%%// The lists will be stored in 'out' as a cell array and can be accessed as out{1}, out{2}, etc.
out = cell(size(A,1)-1,1);
%%// Code that detects the lists using "selective" diagonals
for k = 1:size(A,1)-1
[x,y] = find(triu(A,k).*(~triu(ones(size(A)),k+1)));
out(k) = {[x y]};
end
out(cellfun('isempty',out))=[]; %%// Remove empty lists
%%// Verification - Print out the lists
for k = 1:numel(out)
disp(out{k})
end
Output
1 2
1 3
2 4
1 4
EDIT 1
Basically I will calculate all the the pairwise indices of the matrix to satisfy the criteria set in the question and then simply map them over the given matrix. The part of finding the "valid" indices is obviously the tedious part in it and in this code with some aggressive approach is expensive too when dealing with input matrices of sizes more than 10.
Code
%// Given data, A
A = [0 1 1 1; 1 0 1 1; 1 1 0 1; 1 1 1 0]
%%// Get all pairwise combinations starting with 1
all_combs = sortrows(perms(1:size(A,1)));
all_combs = all_combs(all_combs(:,1)==1,:);
%%// Get the "valid" indices
all_combs_diff = diff(all_combs,1,2);
valid_ind_mat = all_combs(all(all_combs_diff(:,1:2:end)>0,2),:);
valid_ind_mat = valid_ind_mat(all(diff(valid_ind_mat(:,1:2:end),1,2)>0,2),:);
%%// Map the ones of A onto the valid indices to get the lists in a matrix and then cell array
out_cell = mat2cell(valid_ind_mat,repmat(1,[1 size(valid_ind_mat,1)]),repmat(2,[1 size(valid_ind_mat,2)/2]));
A_masked = A(sub2ind(size(A),valid_ind_mat(:,1:2:end),valid_ind_mat(:,2:2:end)));
out_cell(~A_masked)={[]};
%%// Remove empty lists
out_cell(all(cellfun('isempty',out_cell),2),:)=[];
%%// Verification - Print out the lists
disp('Lists =');
for k1 = 1:size(out_cell,1)
disp(strcat(' List',num2str(k1),':'));
for k2 = 1:size(out_cell,2)
if ~isempty(out_cell{k1,k2})
disp(out_cell{k1,k2})
end
end
end
Output
A =
0 1 1 1
1 0 1 1
1 1 0 1
1 1 1 0
Lists =
List1:
1 2
3 4
List2:
1 3
2 4
List3:
1 4
2 3

Answer 2

I'm sure there's a faster way to do it, but here's the obvious solution:
%// Set top half to 0, and find indices of all remaining 1's
A(triu(A)==1) = 0;
[ii,jj] = find(A);
%// Put these in a matrix for further processing
P = [ii jj];
%// Sort indices into 'lists' of the kind you defined
X = repmat({}, size(P,1),1);
for ii = 1:size(P,1)-1
X{ii}{1} = P(ii,:);
for jj = ii+1:size(P,1)
if ~any(ismember(P(ii,:), P(jj,:)))
X{ii}{end+1} = P(jj,:); end
end
end