# Ticket #11312(closed enhancement: fixed)

Opened 2 years ago

## Speed up the computation of the Hilbert basis of a cone

Reported by: Owned by: vbraun mhampton major sage-4.7.2 geometry sd31 novoselt, zaf N/A Andrey Novoseltsev Volker Braun sage-4.7.2.alpha0

My first implementation of the Hilbert basis eventually uses PALP to compute the points in the parallelotope spanned by the rays of a simplicial cone. This can be done much faster with just the Smith normal form of the ray matrix.

This makes it easy to compute the points in the semi-open parallelotope, so the actual number of semigroup generators is sometimes less than the PALP version (which computed the integral points in the closure). As a pleasant side effect, arbitrary dimension cones work now as we are no longer limited to PALP's compile-time bounds.

Apply:

1. trac_11312_speed_up_Hilbert_cone.2.patch

## Change History

### comment:1 Changed 2 years ago by vbraun

• Status changed from new to needs_review
• Description modified (diff)

### comment:2 Changed 2 years ago by novoselt

• Status changed from needs_review to needs_work
• Reviewers set to Andrey Novoseltsev
• Work issues set to non-full-dimensional errors

The speed-up looks impressive! About 20x on trivial examples (no surprise here since we are avoiding system calls to both cdd and PALP), on a "complicated" 6-d example that I picked the new version worked for 15sec and the old one didn't finish before my patience ran out ;-) Am I right that this speed is partially due to taking into account the special structure of the polytopes in which you are computing integral points?

There are, however, issues with non-full-dimensional cones::

sage: Cone([(1,1), (-1,1)], check=False).Hilbert_basis()
(N(1, 1), N(-1, 1), N(0, 1))
sage: Cone([(1,1,0), (-1,1,0)], check=False).Hilbert_basis()
(N(1, 1, 0), N(-1, 1, 0))

### comment:3 Changed 2 years ago by vbraun

I think I'm just implementing the same algorithm as PALP would use to enumerate the points in the parallelotope spanned by the rays, though I'm not sure. Essentially you have to compute the Smith form to enumerate the points, the rest is a simple loop. I'll try to get some more info about Palp's inner workings out of Harald Skarke at the Kreuzer Memorial conference.

And thanks for catching the non-full dimensional cone bug, I should have thought about that but didn't.

### comment:4 Changed 2 years ago by vbraun

• Status changed from needs_work to needs_review

Fixed:

sage: Cone([(1,1), (-1,1)], check=False).Hilbert_basis()
(N(1, 1), N(-1, 1), N(0, 1))
sage: Cone([(1,1,0), (-1,1,0)], check=False).Hilbert_basis()
(N(1, 1, 0), N(-1, 1, 0), N(0, 1, 0))

### comment:5 Changed 2 years ago by vbraun

On further thought, I think its better to also remove non-primitive vectors from the semigroup generators. The new version of the patch saves a few generators, which should speed up the Hilbert basis a little bit.

### comment:6 Changed 2 years ago by vbraun

New version of the patch breaks out the parallelotope_points() function so it can be reused in #11429

### comment:7 Changed 2 years ago by vbraun

• Work issues non-full-dimensional errors deleted

### comment:8 Changed 2 years ago by novoselt

• Status changed from needs_review to needs_work
• Work issues set to One long test fails in cone module

### comment:9 Changed 2 years ago by novoselt

As the patch has to be adjusted one more time anyway - what do you think about creating some kind of sage.geometry.misc module for "helper functions"? Names and rays normalization functions are also natural candidates to go there. This will reduce the potential for circular imports and clarify the structure. cone.parallelotope_points seems a bit strange ;-)

### comment:10 Changed 2 years ago by novoselt

Ah, this particular function is moved to another place in the next patch anyway!

Updated patch

### comment:11 Changed 2 years ago by vbraun

• Status changed from needs_work to needs_review

Good catch! I fixed the offending doctest. Up to the ordering the Hilbert bases before and after the patch are the same, of course.

### comment:12 Changed 2 years ago by novoselt

• Status changed from needs_review to positive_review

All works now!

### Changed 2 years ago by novoselt

Two corrections to REST formatting

### comment:13 Changed 2 years ago by novoselt

• Description modified (diff)
• Work issues One long test fails in cone module deleted

I have made tiny adjustments to the patch to make the documentation compile without warnings, I'll leave it at positive review. (Added one column and replaced a quote with a backward one.)

Volker: subsequent patches may need to be rebased.