Thanks. First, I should note that there must be at least as many values as variables, since each variable must take a distinct value within the set of possible values.

Second, variables that are part of an n-element submatrix need not have n bits set. The sparsest counterexample is:

my @grid = (
    # 0  1  2  3  4  5  
    [ 0, 0, 0, 1, 1, 0, ], 
    [ 0, 0, 0, 1, 0, 1, ], 
    [ 0, 0, 0, 0, 1, 1, ], 
    [ 1, 1, 0, 0, 0, 0, ], 
    [ 1, 0, 1, 0, 0, 0, ], 
    [ 0, 1, 1, 0, 0, 0, ],
);

[download]
.. which can decompose into two 3-element submatrices.

The least sparse version of that is:

my @grid = (
    # 0  1  2  3  4  5  
    [ 0, 0, 0, 1, 1, 1, ], 
    [ 0, 0, 0, 1, 1, 1, ], 
    [ 0, 0, 0, 1, 1, 0, ], 
    [ 1, 1, 1, 1, 1, 1, ], 
    [ 1, 1, 1, 1, 1, 1, ], 
    [ 1, 1, 0, 1, 1, 1, ],
);

[download]
.. which can decompose the same way.

Update: swapped 2 bits in the last row of the sparse matrix, so it actually represents what I'm saying

Hugo

In reply to Re^2: decomposing binary matrices by hv
in thread decomposing binary matrices by hv

Title:
Use:  <p> text here (a paragraph) </p>
and:  <code> code here </code>
to format your post, it's "PerlMonks-approved HTML":


  • Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
  • Titles consisting of a single word are discouraged, and in most cases are disallowed outright.
  • Read Where should I post X? if you're not absolutely sure you're posting in the right place.
  • Please read these before you post! —
  • Posts may use any of the Perl Monks Approved HTML tags:
    a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, details, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, summary, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
  • You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)
            For:     Use:
    & &amp;
    < &lt;
    > &gt;
    [ &#91;
    ] &#93;
  • Link using PerlMonks shortcuts! What shortcuts can I use for linking?
  • See Writeup Formatting Tips and other pages linked from there for more info.