#19197: LatticePoset: add breadth()
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Reporter: jmantysalo | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.9
Component: combinatorics | Resolution:
Keywords: | Merged in:
Authors: Jori Mäntysalo | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
u/jmantysalo/latticeposet__add_breadth__|
cd99bd6b82aaece6bd535d6fdfd01f7bfb28a8c5
Dependencies: | Stopgaps:
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Changes (by ncohen):
* commit: => cd99bd6b82aaece6bd535d6fdfd01f7bfb28a8c5
Comment:
Hello Jori,
Here is a way to improve the algorithm a bit:
Let us say that a set `S` is "locally_minimal" if the join of the elements
in `S` is different from the join of the elements in <any proper subset in
`S`>. What you are looking for is the set `locally_mininal` set of maximum
cardinality.
Why do you iterate on antichains? You iterate on antichains, because you
know that in *any* "locally_minimal" set `S`, any 2-subset of `S` must be
locally minimal. And the sets of locally minimal sets of cardinality 2 are
precisely the antichains.
Why wouldn't you go further? Indeed, the same works for anything greater
than 2: in any locally minimal set, *any* proper subset is also locally
minimal.
Your implementation, however, does not use that. You test all antichains,
but when testing those of size 5 you do not use the information obtained
from those of size 4.
So here is a way out: enumerate all locally minimal sets directly, in
increasing order of size. This can be done with
`subsets_with_hereditary_property` [1], to which you can feed a function
that detects if a given set of points is locally minimal (when you remove
only one element).
This should do the trick, and *use* the information that <some given set
of size 3 is not locally minimal> when trying to figure out one of size 4.
Note: It may be slower for small cases, but better above.
Additionally, your code may spend a lot of time hashing your elements: it
may be better to work directly with the integer ID of the poset's elements
(and to access the matrix directly).
Also, could you be a bit more formal in the definition of 'breadth'? I did
not follow it at first (english is not a well-paenthesized language). It
would also be cool if you could redirect toward a textbook/paper that
defines it: I found it in "Semimodular Lattices Theory and Applications"
where it is said to originate from "Lattice Theory" by Birkhoff.
Nathann
[1]
http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/subsets_hereditary.html
----
New commits:
||[http://git.sagemath.org/sage.git/commit/?id=cd99bd6b82aaece6bd535d6fdfd01f7bfb28a8c5
cd99bd6]||{{{Added function breadth() to lattices.}}}||
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Ticket URL: <http://trac.sagemath.org/ticket/19197#comment:2>
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