Change all but one value to one at random within rows

Suppose I have the following matrix

[ 1 0 1 1 0; 0 1 1 0 0; 1 1 1 0 0; 0 0 0 0 0 ; 0 1 0 0 0]

If a particular row contains two or more ones, I want to change all but one of them to zero

I need the choice to be a random. Thus when implemented, the matrix might become

[ 0 0 0 1 0; 0 1 0 0 0; 1 0 0 0 0; 0 0 0 0 0 ; 0 1 0 0 0]

and a subsequent time

[ 1 0 0 0 0; 0 0 1 0 0; 1 0 0 0 0; 0 0 0 0 0 ; 0 1 0 0 0]

These matrices are large (maybe 2e6 by 10) and speed is an issue. Any help greatly appreciated.

Answer 1

[Robert]

% locate all rows and columns with ones
[a,b] = find(x);

% randomize the order of the rows and keep the column indices aligned
ii = randperm(length(a));
a = a(ii);
b = b(ii);

% keep only the first instance of a row
[a,ii] = unique(a);
b = b(ii);

% assign the ones to our output
if islogical(x)
    y = false(size(x));
else
    y = zeros(size(x),'like',x);
end
ii = sub2ind(size(x),a,b);
y(ii) = 1;

[J T]

Thanks very much!