Matrix of 0s and 1s Where Assignment in Subsequent Rows are Contingent on the Previous Row

Question

I'd like to create a Matrix in MATLAB where:

The First row consists of a random arrangement of 0s and 1s, split evenly (i.e. 50-50).

The Second row randomly assigns zeros to 50% of the 0s and 1s in the first row, and ones to the remaining 50%.

The Third row randomly assigns zeros to 50% of the 0s and 1s in the second row, and ones to the remaining 50%.

Non-randomized Example:

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

Any suggestions?

Answer 1

A solution based on checking whether numbers are bigger or smaller than the median. As long as the number of columns tested is even, exactly half of a set of random doubles will be bigger than the median, and half will be smaller. This guarantees that there's exactly 50% of bits get flipped.

nRows = 3;
nCols = 16; %# divisible by 4

%# seed the array
%# assume that the numbers in each row are unique (very, very likely)
array = rand(nRows,nCols); 


out = false(nRows,nCols);

%# first row is special
out(1,:) = array(1,:) > median(array(1,:));

%# for the rest of the row, check median for the zeros/ones in the previous row
for iRow = 2:nRows
    zeroIdx = out(iRow-1,:) == 0;
    %# > or < do not matter, both will replace zeros/ones 
    %# and replace with exactly half zeros and half ones
    out(iRow,zeroIdx) = array(iRow,zeroIdx) > median(array(iRow,zeroIdx));
    out(iRow,~zeroIdx) = array(iRow,~zeroIdx) > median(array(iRow,~zeroIdx));
end

Answer 2

I'd offer a short bsxfun solution:

%// number of divisions
n = 4;

%// unshuffled matrix like in your example
unshuffled = bsxfun(@(a,b) mod(a,2*b) > b-1, meshgrid(1:n^2,1:n) - 1, (2.^((n-1):-1:0)).') %'

%// shuffle columns
shuffled = unshuffled(:,randperm(n^2))

---

unshuffled =

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


shuffled =

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

First you need to create the unshuffled matrix, which can be done by comparing the matrix generated by meshgrid(1:n^2,1:n) with a row dependent modulus. Finally you just need to shuffle the columns.