MATLAB：生成二进制矩阵的所有可能组合，每列中只有一个“ 1”

Question

MATLAB : I want to find out how to generate all possible combination of a matrix (N by M) where: - elements are "1" and "0". - there should be only one "1" in each column. - no limitation for rows.so multiple "1" is allowed within each row.

one possible state eg N=5 , M=6

1 0 1 0 0 0
0 1 0 0 0 1
0 0 0 0 0 0
0 0 0 1 1 0
0 0 0 0 0 0

furthermore I want to generate each possible combination matrix , then compute something (eg a utility function in my problem) like this :

generate one possible matrix C
.
  .
    for i=1:N
      for j=1:M
        do something on C(:,:)
      end
    end
  .
.

(in the manner of exhaustive search)

Answer 1

There will be lots . Usually when a question here involves: How do I generate all possible X , the real answer is: Don't do it, there are too many possible X . Look for a different approach to your problem.

Nevertheless, you could use the number representation in the base of your number of rows:

Approach using: dec2base

Disclaimer: Due to limitations of dec2base this will only work for 2<=rows<=36 (Which is hopefully enough. Otherwise we would copy-edit the file dec2base.m and remove it's two last lines and the error check in line 24 to achieve arbitrary values of 2<=rows . This code I won't post for copyright reasons.).

rows = 5; 
cols = 6;
assert((2<=rows)&&(rows<=36),'The dec2base-approach will only work for 2<=rows<=36');
symbols = dec2base(0:rows-1, rows, 1);
for ii = 0:rows^cols-1
    % Compute ii in base rows.
    iibR = dec2base(ii, rows, cols);
    C = bsxfun(@eq, symbols, iibR);
    disp(C);
end

Generating the tuples without dec2base :

We could also generate these tuples that represent our numbers using ndgrid .

%%// Data
rows = 3;
cols = 4;
%%// Compute all k-tuples of numbers 1:n
n = rows;
k = cols;
Cs = cell(1,k);
[Cs{:}] = ndgrid(1:n);
tuples = reshape(cat(n+1, Cs{:}),n^k,[]);
%%// Compute matrices
for ii = 1:size(tuples,1);
    C = bsxfun(@eq, (1:rows).', tuples(ii,:));
    disp(C);
end