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Topic: how to extract the structure....
Replies: 1   Last Post: Mar 24, 2013 2:24 AM

 James Waldby Posts: 545 Registered: 1/27/11
Re: how to extract the structure....
Posted: Mar 24, 2013 2:24 AM

On Sat, 23 Mar 2013 11:22:57 -0700, Dieter von Holten wrote:
> lets assume i have a set of lists (4 letter words...), like below:

> (1) abcd, aabb, abda, dbdb, bbca, afah, ...
> when i swap two letters, like a<->b, i get another set of lists:
> or, by swapping a<->c in (1):
> (3) cbad, ccbb, cbdc, dbdb, bbac, cfch, ...
> the lists look different, but they share the same structure
> (4) 1234, 1122, 1241, 4242, 2231, 1516, ...

...[snip other examples]...
> the letters have no (implicit) value, they are just different symbols.
>
> any ideas ? what shall i google?

It might be that this problem can be put into correspondence
with the Graph isomorphism problem, whose complexity of solving
is not known, but is an important open question. See
<http://en.wikipedia.org/wiki/Graph_isomorphism_problem>.

Or, this problem might be easier for average cases. I didn't
understand examples 5 and 6 (snipped for brevity) but suppose
the word lists were related by word-reordering and simple
alphabetic substitution. If so, testing for isomorphism probably
can be done quickly by canonicalizing each word list. Eg,
replace the most common character by 1, the next most common
by 2, etc. When a tie occurs, break it by finding the most
common first character, or if that ties, the most common 2nd
character, or most common in the next combination of columns
to be considered, etc. As characters are named in this way,
remove them from future counting. This approach serves to
canonicalize examples 1-4 but presumably there are examples
it doesn't work for.

--
jiw