Reimplement IntegerLists using Polyhedron.integral_points()

[jdemeyer]

This fixes #17548.

It also adds new features to IntegerLists:

1. Negative integers are allowed (but the default still is min_part=0).

2. There does not need to be a fixed sum, one can do for example IntegerLists(max_part=2) for all lists of integers <= 2. One can also give a lower/upper bound for the sum.

Note that the current implementation requires, for a given length, that there are only finitely many lists of that length. This limitation could be lifted in the future.

[jdemeyer]

The Sage MILP solvers cannot enumerate all solutions => closing as invalid.

[ncohen]

I do not understand what this is... Did you copy/paste the original file? It seems that you copy/pasted the original files and made some modifications to it O_o

Nathann

---

New commits:

​492696f

Reimplement IntegerLists using Polyhedron.integral_points()

[jdemeyer]

Yes, that's what I did. Anyway, this is still very much work in progress...

[ncohen]

Yes, that's what I did. Anyway, this is still very much work in progress...

Oh, okay!

Nathann

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​eb6ba85	Fix integral_points() for polyhedra in 0 dimensions
​5896b11	Reimplement IntegerLists using Polyhedron.integral_points()

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​c835117

Reimplement IntegerLists using Polyhedron.integral_points()

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​755b67a

Reimplement IntegerLists using Polyhedron.integral_points()

[jdemeyer]

This now seems to work reasonably well. Not yet ready, but good enough for example to compare with the existing implementation. That's how I found all the bugs at #17548.

Due to the polyhedra overhead, it is generally (a lot) slower than the existing code.

[ncohen]

Replying to jdemeyer:

This now seems to work reasonably well. Not yet ready, but good enough for example to compare with the existing implementation. That's how I found all the bugs at #17548.

Due to the polyhedra overhead, it is generally (a lot) slower than the existing code.

Is it worth changing this branch so that it changes the existing code instead of adding a new file ? As you said: let's be correct first, *then* fast.

Nathann

[jdemeyer]

My code does not yet support all (undocumented!) features of the old IntegerListsLex, so we cannot yet replace IntegerListsLex.

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​674a518

Reimplement IntegerLists using Polyhedron.integral_points()

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​4e5dbae

Reimplement IntegerLists using Polyhedron.integral_points()

[jdemeyer]

This now implements Compositions using my new IntegerListsLex (is the lex ordering really important? I guess not, it's for sure nowhere documented). However, doing the same for Partitions leads to all kinds of breakage and I don't understand why.

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​c0772f8

Reimplement IntegerLists using Polyhedron.integral_points()

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​631c476

Reimplement IntegerLists using Polyhedron.integral_points()

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​3b7f4cd

Reimplement IntegerLists using Polyhedron.integral_points()

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​936a2c7

Reimplement IntegerLists using Polyhedron.integral_points()

[jdemeyer]

Note that this changes 3 tests (just reordering the output) in src/sage/tests/book_schilling_zabrocki_kschur_primer.py

[jdemeyer]

The code on this ticket is essentially complete, I just need to add more doctests to comply with the "coverage" policy.

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​c9fa1c4

Reimplement IntegerLists using Polyhedron.integral_points()

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​b0a04aa

Reimplement IntegerLists using Polyhedron.integral_points()

[tscrim]

Do you have some timings?

Also, both of these are wrong:

sage: Compositions(3, max_length=2, inner=[1,1,1]).list()
[]
sage: Compositions(10, outer=[4], inner=[1,1]).list()
[]

The first should be [[2, 1], [1, 2]] since the inner (or outer) are not related to the min or max lengths. For the second, the inner composition is extendedby the minimum part, so there are many such compositions, such as [4,6], [2,8], etc.

[vdelecroix]

Replying to tscrim:

Do you have some timings?

Slow is better than wrong, isn't it? ;-)

[jdemeyer]

Replying to tscrim:

Do you have some timings?

My code is much slower.

Also, both of these are wrong:

sage: Compositions(3, max_length=2, inner=[1,1,1]).list()
[]
sage: Compositions(10, outer=[4], inner=[1,1]).list()
[]

The first should be [[2, 1], [1, 2]]

I don't think so. The Compositions code explicitly adds the length of the inner argument as minimal length. I didn't change this.

[jdemeyer]

Setting to blocker status since either #17920 or #17956 should be fixed.

[jdemeyer]

Replying to tscrim:

The first should be [[2, 1], [1, 2]] since the inner (or outer) are not related to the min or max lengths.

To clarify: you might be confusing with the floor and ceiling arguments of IntegerListsLex. Those do not have any effect on the length, but inner/outer do add lower/upper bounds to the length. With both the existing code as well as with my code, we have for example

sage: Compositions(3, inner=[1,1,1]).list()
[[1, 1, 1]]

[tscrim]

On the ordering, from the title of the class, the output should be in lexicographical order. Moreover, since these are EnumeratedSets, the change in the ordering could lead to subtle changes that breaks people's code.

Compositions also does a similar thing with the max length and outer, so I agree that those should be empty. However IMO these tests should be in Compositions.

@vdelecroix It's mostly correct and there is a lot of code which depends on this being fast. Minimal slowdowns are okay (IMO), but significant slowdowns are unacceptable.

[ncohen]

On the ordering, from the title of the class, the output should be in lexicographical order. Moreover, since these are EnumeratedSets, the change in the ordering could lead to subtle changes that breaks people's code.

We can sort it before returning it I guess.

@vdelecroix It's mostly correct

Jeroen compiled many related bugs in the description of #17548.

Minimal slowdowns are okay (IMO), but significant slowdowns are unacceptable.

Significant slowdown can be a problem, that's for sure. If they turn out to be our only way to have a code which does not return wrong results, however, we will learn to live with them.

Nathann

[aschilling]

I think it is great that Jeroen implemented this to get the correct results! Of course we do want fast code at the end.

As I mentioned on sage-devel, the order of lists of tableaux does not matter very much.

As Travis mentioned, there might be some subtle places where the order matters. One example that comes to mind is that the representations of S_n and characters are returned as matrices with rows and columns indexed by integers instead of partitions. So if the order of partitions changes, the interpretation of the results might change!

[jdemeyer]

Replying to tscrim:

On the ordering, from the title of the class, the output should be in lexicographical order.

The name is now IntegerLists and I do provide IntegerListsLex for "backwards compatibility" which does sort.

Since the documentation of neither Partitions nor Compositions says anything about the order, I think it's allowed to change the order.

[jdemeyer]

About the speed: if you manage to fix all existing bugs in the IntegerListsLex code, you can again use that implementation for Compositions and Partitions. It's probably good to have two indepdendent implementations anyway.

Even better would of course be that somebody speeds up Polyhedron().integral_points() which would benefit everybody using polyhedra.

[nthiery]

Dear Jeroen,

Thanks a lot for taking action! It's definitely a good thing to have a good connection between IntegerListLex? and Polyhedron, as there is some non trivial overlap. The main differences is that IntegerListLex? was specifically designed for allowing for (essentially) Constant Amortized Time lexicographic Iteration in almost constant memory, which is an important feature.

So I can see a work path along the following lines:

• Get this ticket in to have a robust implementation of list

• Completely rewrite the current IntegerListLex? iterator to be robust (it's definitely possible); keep the Polyhedron implementation for testing purposes as well as for counting, ...

• Optimize the iterator (Cythonization, using ClonableIntArray?, ...).

Please do not change the enumeration order, at least as default: quite some code depends on it (I agree, this should be made explicit in the documentation). The proposed generalizations (n in a range, negative entries) are fine since the iterator could be made to handle them.

Cheers,

Nicolas

[jdemeyer]

Replying to nthiery:

Please do not change the enumeration order, at least as default

I disagree with this: the default should be "do not sort, return stuff in the fastest possible way". Sorting an iterator is very expensive and should only be done if really needed.

quite some code depends on it

Is that really true? The only doctest failures that I saw where "obvious" failures where some list order changed, I didn't see anything subtle.

[jdemeyer]

Replying to nthiery:

keep the Polyhedron implementation for testing purposes as well as for counting, ...

I'm not sure about the counting... I guess a well-written Cython implementation of IntegerListsLex will usually be faster than the current polyhedra code. Profiling shows that a lot of time is spent in just constructing the polyhedra (if there are not so many points, enumerating them takes a lot less time than constructing the polyhedron in the first place).

[ncohen]

Hello,

I'm not sure about the counting... I guess a well-written Cython implementation of IntegerListsLex will usually be faster than the current polyhedra code.

True. This being said, your polyhedron code may very well be 'all we can do' to implement this feature while handling all possible combinations of parameters.

Profiling shows that a lot of time is spent in just constructing the polyhedra

True. Do you have any idea where that comes from ? I had similar troubles with the Poset constructor (related to memory usage).

Nathann

[jdemeyer]

Replying to ncohen:

Profiling shows that a lot of time is spent in just constructing the polyhedra

True. Do you have any idea where that comes from ?

It's just PPL.

Interestingly, arithmetic with infinity also shows up quite high in the profiling reports (up to 10% of the time), so optimizing src/sage/rings/infinity.py will also give some speedup.

[ncohen]

Interestingly, arithmetic with infinity also shows up quite high in the profiling reports (up to 10% of the time), so optimizing src/sage/rings/infinity.py will also give some speedup.

Aahahah. Yaeh, Vincent has been fighting a lot with some abstractions we have, which makes code run *much* slower. For +oo he advises to solve the problem by using float("inf") instead of Infinity. It is *MUCH* faster.

At some point he got some crazy somewhere by adding 'from math import sqrt' in a module to overwrite Sage's sqrt function.

Nathann

[jdemeyer]

Replying to ncohen:

Interestingly, arithmetic with infinity also shows up quite high in the profiling reports (up to 10% of the time), so optimizing src/sage/rings/infinity.py will also give some speedup.

Aahahah. Yaeh, Vincent has been fighting a lot with some abstractions we have, which makes code run *much* slower. For +oo he advises to solve the problem by using float("inf") instead of Infinity.

In general, I don't like the "X is slow, therefore let's not use X" mentality. My idea is: "let's use X and then optimize X".

[mmezzarobba]

Replying to jdemeyer:

Interestingly, arithmetic with infinity also shows up quite high in the profiling reports (up to 10% of the time), so optimizing src/sage/rings/infinity.py will also give some speedup.

Aahahah. Yaeh, Vincent has been fighting a lot with some abstractions we have, which makes code run *much* slower. For +oo he advises to solve the problem by using float("inf") instead of Infinity.

In general, I don't like the "X is slow, therefore let's not use X" mentality. My idea is: "let's use X and then optimize X".

On a tangential note: if someone makes major changes to the infinity rings, please consider adding a NaN element to them.

[ncohen]

In general, I don't like the "X is slow, therefore let's not use X" mentality. My idea is: "let's use X and then optimize X".

Well, for the sqrt problem we were only computing on floats, and sqrt(5) in Sage is not a float. Having this symbolic value in the code we compared it with floats and this had a cost we had no reason to pay, so from math import sqrt made total sense, and there was nothing in Sage's sqrt that we could have wanted to change.

As per Sage's Infinity... Well, it also seems to have been designed with a different aim in mind. I usually want speed, and I do not want to pay for pointless abstraction. In infinity.py you will find parents, elements, rings and generators, while the feature I need is already provided by the constant LONG_MAX.

This Sage object is called 'infinity', but it turns out that one definition of "infinity" cannot cover all uses that we have for infinity on a computer. And I don't think that we could beat a single CPU instruction while dealing with parents and elements in a .py file.

Nathann

[jdemeyer]

Replying to ncohen:

while the feature I need is already provided by the constant LONG_MAX.

In this case, we shouldn't aritificially restrict to longs:

sage: IntegerLists(10^100, max_length=1).list()
[[10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]]

This Sage object is called 'infinity', but it turns out that one definition of "infinity" cannot cover all uses that we have for infinity on a computer.

I think we can have one definition which covers all uses within Sage. There is nothing fundamentally wrong with Infinity, it's just slow.

[jdemeyer]

Note the difference of a factor 20 between the following:

sage: b = ZZ(1); a = Infinity; timeit('a < b', repeat=20, number=10^4)
10000 loops, best of 20: 11.7 µs per loop
sage: b = ZZ(1); a = QQ(2); timeit('a < b', repeat=20, number=10^4)
10000 loops, best of 20: 681 ns per loop

A proper Cython implementation of Infinity should match the speed for QQ.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​5f81a0a

Move doctests

[jdemeyer]

Replying to tscrim:

However IMO these tests should be in Compositions.

Done

[nthiery]

Replying to jdemeyer:

Replying to nthiery:

Please do not change the enumeration order, at least as default

I disagree with this: the default should be "do not sort, return stuff in the fastest possible way". Sorting an iterator is very expensive and should only be done if really needed.

For IntegerList? itself, any default is fine, and sorting is certainly very bad. But for Partitions, Compositions, ... users are really expecting lexicographic order. Besides, this is only a temporary solution, and we will switch back to lexicographic once we have a proper implementation of IntegerListLex?.

Is that really true? The only doctest failures that I saw where "obvious" failures where some list order changed, I didn't see anything subtle.

That's a point indeed; my feeling is that we have been lucky, although it could be that some changes w.r.t. MuPAD-Combinat makes the code less dependent on that feature. I am worried by user's personal code out there. Well, if you are willing to help those guys ...

Cheers,

Nicolas

[nthiery]

Replying to jdemeyer:

I'm not sure about the counting... I guess a well-written Cython implementation of IntegerListsLex will usually be faster than the current polyhedra code. Profiling shows that a lot of time is spent in just constructing the polyhedra (if there are not so many points, enumerating them takes a lot less time than constructing the polyhedron in the first place).

Agreed, especially if we further go parallel: counting through polyhedral methods only becomes relevant for relatively large polyhedron. But this would be a very useful feature. So count would eventually have some threshold to choose between the two methods.

By the way: we don't yet use Barvinok-like algorithms for counting (e.g. through LattE), or do we? This could make a difference too.

Cheers,

Nicolas

[jdemeyer]

Replying to nthiery:

By the way: we don't yet use Barvinok-like algorithms for counting (e.g. through LattE), or do we? This could make a difference too.

I just read the first paragraph of the LattE manual and it does exactly what we need for counting:

1.1    What is LattE?

The name “LattE” is an abbreviation for “Lattice point Enumeration.” LattE
was developed in 2001 to count lattice points contained in convex polyhedra
defined by linear equations and inequalities with integer coefficients. The poly-
hedra can be of any (reasonably small) dimension.

[jdemeyer]

Replying to nthiery:

But for Partitions, Compositions, ... users are really expecting lexicographic order.

Well, certainly for Compositions, the current order is not defined:

sage: Compositions(2).list()
[[1, 1], [2]]
sage: Compositions(2, max_slope=0).list()
[[2], [1, 1]]

[ncohen]

​http://trac.sagemath.org/ticket/17529#comment:11 (we already have a LattE package)

[nthiery]

Replying to jdemeyer:

I just read the first paragraph of the LattE manual and it does exactly what we need for counting:

Yes indeed! See also #15180.

Btw: in case this could be useful, the developers are in Davis where I currently am for the upcoming Sage Days 64.

Cheers,

Nicolas

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​6f31649

Add two extra tests

[tscrim]

FYI - float('inf') works quite well as a good and fast substitute for Infinity (which is why it is used in the current IntegerListsLex).

[jdemeyer]

Replying to nthiery:

But for Partitions users are really expecting lexicographic order.

For which applications does this really matter?

I know that's how partitions are traditionally written down and how things are done in Sage historically. But I don't think that's enough reason to not change it, especially given the fact that the documentation doesn't say anything about the order. Any order of the list of partitions is a good answer mathematically.

[tscrim]

I'm worried this could lead to errors being raised when trying to convert between different bases of the symmetric functions (which are indexed by partitions). IIRC the code relies on some of the (graded) transition matrices being upper triangular, which requires the order be compatible with dominance ordering.

[jdemeyer]

Replying to tscrim:

I'm worried this could lead to errors being raised when trying to convert between different bases of the symmetric functions (which are indexed by partitions). IIRC the code relies on some of the (graded) transition matrices being upper triangular, which requires the order be compatible with dominance ordering.

To be honest, I don't know what you mean mathematically. But, like I said, the fact that there are no strange doctest failures shows that the issue cannot be so serious.

And in the cases where the order really matters, I think those places should simply explicitly sort or use IntegerListsLex. Slowing down all of Paritions() just because one or two applications require it seems stupid.

[aschilling]

But, like I said, the fact that there are no strange doctest failures shows that the issue cannot be so serious.

This might be an issue though

sage: S=SymmetricGroup(3)
sage: S.character_table()
[ 1 -1  1]
[ 2  0 -1]
[ 1  1  1]
sage: type(S.character_table())
<type 'sage.matrix.matrix_cyclo_dense.Matrix_cyclo_dense'>

The character table should really be indexed by partitions (or conjugacy classes of S_n). So the meaning of the matrix changes if the order of the list of partitions changes.

[jdemeyer]

Replying to aschilling:

This might be an issue though

sage: S=SymmetricGroup(3)
sage: S.character_table()
[ 1 -1  1]
[ 2  0 -1]
[ 1  1  1]
sage: type(S.character_table())
<type 'sage.matrix.matrix_cyclo_dense.Matrix_cyclo_dense'>

The character table should really be indexed by partitions (or conjugacy classes of S_n). So the meaning of the matrix changes if the order of the list of partitions changes.

Okay, so the meaning of the matrix changes. There is nothing wrong with output changing as long as things stay mathematically correct and internally consistent.

[dimpase]

Replying to jdemeyer:

Replying to nthiery:

By the way: we don't yet use Barvinok-like algorithms for counting (e.g. through LattE), or do we? This could make a difference too.

I just read the first paragraph of the LattE manual and it does exactly what we need for counting:

Moreover, counting the points is faster than enumerating the points; there could be exponentially many points to count, but still the number of them is only polynomial in the input size, for fixed dimension, and LattE provides a truly polynomial-time procedure for this counting.

[chapoton]

a badly formated doc in composition.py

     TESTS::
 
+    Check that :trac:`17548` is fixed::

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​62b56ba

Improve IntegerLists_polyhedron; sort Partitions by default

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​5096e5f

Fix documentation

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​c9dce18	Remove sig_on() from __dealloc__
​621d467	Merge commit 'c9dce18385fd59755cbcced5f1804bf60b19df07' into t/17920/ticket/17920

[vbraun]

Whats your plan with the code here? It might be useful to check. Thoughts?

[tscrim]

This code is useful as it works in more general context than the new IntegerListsLex as it allows entries in ZZ (instead of just NN) and when lex ordering doesn't hit every element in finite time. Plus I like alternative algorithms for testing, and this is faster in certain cases currently as well. I just need to find some time to review this.

[nthiery]

Indeed, we want this code in Sage! I promised Jeroen I would rebase his code, and work on the review. This could be a good sprint for Sage Days 67. But yes this certainly is not a blocker anymore for 6.6.

[jdemeyer]

Needs to be rebased in any case. I might also try to make it work in all cases of infinite iterator (currently, I still require that there are only finitely many lists of every given length).

[jdemeyer]

I would actually like to redesign IntegerListsLex and IntegerLists_polyhedron such that they share code: they could be the same class but with a different implementation for __iter__ and __contains__.

[vdelecroix]

Replying to jdemeyer:

I would actually like to redesign IntegerListsLex and IntegerLists_polyhedron such that they share code: they could be the same class but with a different implementation for __iter__ and __contains__.

+1

why not several iterators on the same class? (with a reasonable one by default in __iter__).

[jdemeyer]

Replying to vdelecroix:

Replying to jdemeyer:

I would actually like to redesign IntegerListsLex and IntegerLists_polyhedron such that they share code: they could be the same class but with a different implementation for __iter__ and __contains__.

+1

why not several iterators on the same class? (with a reasonable one by default in __iter__).

Well, it will be something along those lines. But I haven't thought too much about the actual design.

[jdemeyer]

Replying to nthiery:

I promised Jeroen I would rebase his code

Thanks for the offer, but that will not be needed (in any case, it's just merging with -X theirs essentially)

[nthiery]

Replying to jdemeyer:

Replying to vdelecroix:

Replying to jdemeyer:

I would actually like to redesign IntegerListsLex and IntegerLists_polyhedron such that they share code: they could be the same class but with a different implementation for __iter__ and __contains__.

+1

why not several iterators on the same class? (with a reasonable one by default in __iter__).

Well, it will be something along those lines. But I haven't thought too much about the actual design.

+1 to sharing code between the classes; in fact I had put a mental note on doing this when I offered to do the merge :-)

Having a class that can handle any set of constraints, even if it does only containment check, would be useful. We would need this in particular to properly refactor integer vectors.

I am not sure whether we want a single class, or two classes:

class IntegerLists:
    __iter__ -> __iter__ on the polyhedron

class IntegerListsLex(IntegerLists):

With the second one having the additional specification that the enumeration shall be lexicographic.

Thoughts?

Cheers,

Nicolas