Problem 885. Create logical matrix with a specific row and column sums

Created by Amro

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Given two numbers <tt>n</tt> and <tt>s</tt>, build an <tt>n-by-n</tt> logical matrix (of only zeros and ones), such that both the row sums and the column sums are all equal to <tt>s</tt>. Additionally, the main diagonal must be all zeros.You can assume that: <tt>0 &lt; s &lt; n</tt>&nbsp;Example:Take <tt>n=10</tt> and <tt>s=3</tt>, here is a possible solution<pre class="language-matlab">M =
 0 1 0 0 1 1 0 0 0 0
 0 0 1 0 1 1 0 0 0 0
 0 0 0 0 1 1 0 0 1 0
 0 0 0 0 0 0 1 1 0 1
 1 0 0 0 0 0 1 0 1 0
 0 1 1 0 0 0 0 1 0 0
 1 0 0 1 0 0 0 0 0 1
 0 0 0 1 0 0 0 0 1 1
 1 1 0 0 0 0 0 1 0 0
 0 0 1 1 0 0 1 0 0 0
</pre>Note that the following conditions are all true:<pre class="language-matlab">all(sum(M,1)==3) % column sums equal to s
all(sum(M,2)==3) % row sums equal to s
all(diag(M)==0) % zeros on the diagonal
islogical(M) % logical matrix
ndims(M)==2 % 2D matrix
all(size(M)==n) % square matrix
</pre>&nbsp;Unscored bonus:Visualize the result as a graph where <tt>M</tt> represents the adjacency matrix:<pre class="language-matlab">% circular layout
t = linspace(0, 2*pi, n+1)';
xy = [cos(t(1:end-1)) sin(t(1:end-1))];
subplot(121), spy(M)
subplot(122), gplot(M, xy, '*-'), axis image
</pre>

Given two numbers *|n|* and *|s|*, build an |n-by-n| logical matrix (of only zeros and ones), such that both the row sums and the column sums are all equal to |s|. Additionally, the main diagonal must be all zeros.

You can assume that: |0 < s < n|

&nbsp;

*Example:*

Take |n=10| and |s=3|, here is a possible solution

M =
       0     1     0     0     1     1     0     0     0     0
       0     0     1     0     1     1     0     0     0     0
       0     0     0     0     1     1     0     0     1     0
       0     0     0     0     0     0     1     1     0     1
       1     0     0     0     0     0     1     0     1     0
       0     1     1     0     0     0     0     1     0     0
       1     0     0     1     0     0     0     0     0     1
       0     0     0     1     0     0     0     0     1     1
       1     1     0     0     0     0     0     1     0     0
       0     0     1     1     0     0     1     0     0     0

Note that the following conditions are all true:

all(sum(M,1)==3)     % column sums equal to s
  all(sum(M,2)==3)     % row sums equal to s
  all(diag(M)==0)      % zeros on the diagonal
  islogical(M)         % logical matrix
  ndims(M)==2          % 2D matrix
  all(size(M)==n)      % square matrix

&nbsp;

*Unscored bonus:*

Visualize the result as a graph where |M| represents the adjacency matrix:

% circular layout
  t = linspace(0, 2*pi, n+1)';
  xy = [cos(t(1:end-1)) sin(t(1:end-1))];
  subplot(121), spy(M)
  subplot(122), gplot(M, xy, '*-'), axis image

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Last Solution submitted on May 18, 2023

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2 Comments

Clément E. on 30 May 2018

Any clue? Does this problem require great mathematics abilities ?

Rafael S.T. Vieira on 23 Jul 2020

Not really, imagine in the simplest case that we could use the diagonals (ignoring this constraint), what the solution would look like? A chessboard pattern would solve it, wouldn't it? Now, how can we work around the constraint?

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