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Trac #29198: Set up hypercube with both Vrep and Hrep (if backend sup…
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…ports it)

Currently, the hypercube is set up with the vertices. This is slow, as
the vertices grow exponentially with dimension:

{{{
sage: %time _ = polytopes.hypercube(8)
CPU times: user 58.6 ms, sys: 0 ns, total: 58.6 ms
Wall time: 58.2 ms
sage: %time _ = polytopes.hypercube(14)
CPU times: user 2.74 s, sys: 19.1 ms, total: 2.76 s
Wall time: 2.76 s
}}}

With #28880 at hand, we can set up the hypercube with both Vrep and
Hrep. If the backend supports it (as in `backend='field'`), then the
double description does not need to be computed. If the backend does not
support it (as in `backend='ppl'`), then the hypercube is set up from
the inequalities, which is much faster:

{{{
sage: %time _ = polytopes.hypercube(8)   # uses ppl
CPU times: user 47.8 ms, sys: 3.19 ms, total: 51 ms
Wall time: 50 ms
sage: %time _ = polytopes.hypercube(14)  # uses ppl
CPU times: user 421 ms, sys: 4.7 ms, total: 426 ms
Wall time: 425 ms
sage: %time _ = polytopes.hypercube(14, backend='field')  # uses both
descriptions
CPU times: user 168 ms, sys: 124 µs, total: 169 ms
Wall time: 168 ms
}}}

URL: https://trac.sagemath.org/29198
Reported by: gh-kliem
Ticket author(s): Jonathan Kliem
Reviewer(s): Jean-Philippe Labbé, Sébastien Labbé
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Release Manager committed Mar 7, 2020
2 parents 1c7ea20 + 91ae9ba commit b9dd82b
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Showing 10 changed files with 238 additions and 172 deletions.
12 changes: 6 additions & 6 deletions src/sage/geometry/integral_points.pyx
View file
Original file line number Diff line number Diff line change
Expand Up @@ -507,19 +507,19 @@ cpdef rectangular_box_points(list box_min, list box_max,
sage: cube = polytopes.cube()
sage: cube.Hrepresentation(0)
An inequality (0, 0, -1) x + 1 >= 0
An inequality (-1, 0, 0) x + 1 >= 0
sage: cube.Hrepresentation(1)
An inequality (0, -1, 0) x + 1 >= 0
sage: cube.Hrepresentation(2)
An inequality (-1, 0, 0) x + 1 >= 0
An inequality (0, 0, -1) x + 1 >= 0
sage: rectangular_box_points([0]*3, [1]*3, cube, return_saturated=True)
(((0, 0, 0), frozenset()),
((0, 0, 1), frozenset({0})),
((0, 0, 1), frozenset({2})),
((0, 1, 0), frozenset({1})),
((0, 1, 1), frozenset({0, 1})),
((1, 0, 0), frozenset({2})),
((0, 1, 1), frozenset({1, 2})),
((1, 0, 0), frozenset({0})),
((1, 0, 1), frozenset({0, 2})),
((1, 1, 0), frozenset({1, 2})),
((1, 1, 0), frozenset({0, 1})),
((1, 1, 1), frozenset({0, 1, 2})))
TESTS:
Expand Down
98 changes: 52 additions & 46 deletions src/sage/geometry/polyhedron/base.py
View file
Original file line number Diff line number Diff line change
Expand Up @@ -1430,7 +1430,7 @@ def Hrepresentation(self, index=None):
sage: p = polytopes.hypercube(3)
sage: p.Hrepresentation(0)
An inequality (0, 0, -1) x + 1 >= 0
An inequality (-1, 0, 0) x + 1 >= 0
sage: p.Hrepresentation(0) == p.Hrepresentation() [0]
True
"""
Expand Down Expand Up @@ -1522,7 +1522,7 @@ def Hrepresentation_str(self, separator='\n', latex=False, style='>=', align=Non
sage: c = polytopes.cube()
sage: c.Hrepresentation_str(separator=', ', style='positive')
'1 >= x2, 1 >= x1, 1 >= x0, x0 + 1 >= 0, x2 + 1 >= 0, x1 + 1 >= 0'
'1 >= x0, 1 >= x1, 1 >= x2, x0 + 1 >= 0, x2 + 1 >= 0, x1 + 1 >= 0'
"""
pretty_hs = [h.repr_pretty(split=True, latex=latex, style=style, **kwds) for h in self.Hrepresentation()]
shift = any(pretty_h[2].startswith('-') for pretty_h in pretty_hs)
Expand Down Expand Up @@ -1574,7 +1574,7 @@ def Hrep_generator(self):
sage: p = polytopes.hypercube(3)
sage: next(p.Hrep_generator())
An inequality (0, 0, -1) x + 1 >= 0
An inequality (-1, 0, 0) x + 1 >= 0
"""
for H in self.Hrepresentation():
yield H
Expand Down Expand Up @@ -1993,8 +1993,8 @@ def an_affine_basis(self):
sage: P.an_affine_basis()
[A vertex at (-1, -1, -1),
A vertex at (1, -1, -1),
A vertex at (-1, -1, 1),
A vertex at (-1, 1, -1)]
A vertex at (1, -1, 1),
A vertex at (1, 1, -1)]
sage: P = polytopes.permutahedron(5)
sage: P.an_affine_basis()
Expand Down Expand Up @@ -2804,7 +2804,7 @@ def a_maximal_chain(self):
A 2-dimensional face of a Polyhedron in ZZ^3 defined as the convex hull of 4 vertices,
A 3-dimensional face of a Polyhedron in ZZ^3 defined as the convex hull of 8 vertices]
sage: [face.ambient_V_indices() for face in chain]
[(), (0,), (0, 4), (0, 1, 4, 5), (0, 1, 2, 3, 4, 5, 6, 7)]
[(), (5,), (0, 5), (0, 3, 4, 5), (0, 1, 2, 3, 4, 5, 6, 7)]
TESTS::
Expand Down Expand Up @@ -3333,14 +3333,14 @@ def is_prism(self, certificate=False):
True
sage: P.is_prism(certificate=True)
(True,
[[A vertex at (-1, -1, 1),
A vertex at (-1, 1, 1),
A vertex at (1, -1, 1),
A vertex at (1, 1, 1)],
[A vertex at (-1, -1, -1),
[[A vertex at (1, -1, -1),
A vertex at (1, 1, -1),
A vertex at (1, 1, 1),
A vertex at (1, -1, 1)],
[A vertex at (-1, -1, 1),
A vertex at (-1, -1, -1),
A vertex at (-1, 1, -1),
A vertex at (1, -1, -1),
A vertex at (1, 1, -1)]])
A vertex at (-1, 1, 1)]])
sage: Q = polytopes.cyclic_polytope(3,8)
sage: Q.is_prism()
False
Expand Down Expand Up @@ -3740,13 +3740,15 @@ def triangulate(self, engine='auto', connected=True, fine=False, regular=None, s
sage: triangulation = cube.triangulate(
....: engine='internal') # to make doctest independent of TOPCOM
sage: triangulation
(<0,1,2,7>, <0,1,4,7>, <0,2,4,7>, <1,2,3,7>, <1,4,5,7>, <2,4,6,7>)
(<0,1,2,7>, <0,1,5,7>, <0,2,3,7>, <0,3,4,7>, <0,4,5,7>, <1,5,6,7>)
sage: simplex_indices = triangulation[0]; simplex_indices
(0, 1, 2, 7)
sage: simplex_vertices = [ cube.Vrepresentation(i) for i in simplex_indices ]
sage: simplex_vertices
[A vertex at (-1, -1, -1), A vertex at (-1, -1, 1),
A vertex at (-1, 1, -1), A vertex at (1, 1, 1)]
[A vertex at (1, -1, -1),
A vertex at (1, 1, -1),
A vertex at (1, 1, 1),
A vertex at (-1, 1, 1)]
sage: Polyhedron(simplex_vertices)
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 4 vertices
Expand Down Expand Up @@ -4860,13 +4862,13 @@ def face_truncation(self, face, linear_coefficients=None, cut_frac=None):
sage: edge_trunc.f_vector()
(1, 10, 15, 7, 1)
sage: tuple(f.ambient_V_indices() for f in edge_trunc.faces(2))
((0, 4, 5, 6),
(0, 1, 6, 7),
(5, 6, 7, 8, 9),
(3, 4, 5, 9),
(1, 2, 7, 8),
(2, 3, 8, 9),
(0, 1, 2, 3, 4))
((0, 5, 6, 7),
(1, 4, 5, 6, 8),
(6, 7, 8, 9),
(0, 2, 3, 7, 9),
(1, 2, 8, 9),
(0, 3, 4, 5),
(1, 2, 3, 4))
sage: face_trunc = Cube.face_truncation(Cube.faces(2)[2])
sage: face_trunc.vertices()
(A vertex at (1, -1, -1),
Expand Down Expand Up @@ -5840,23 +5842,27 @@ def faces(self, face_dimension):
sage: p = polytopes.hypercube(4)
sage: list(f.ambient_V_indices() for f in p.faces(3))
[(0, 2, 4, 6, 8, 10, 12, 14),
(0, 1, 4, 5, 8, 9, 12, 13),
(0, 1, 2, 3, 8, 9, 10, 11),
(0, 1, 2, 3, 4, 5, 6, 7),
(1, 3, 5, 7, 9, 11, 13, 15),
(2, 3, 6, 7, 10, 11, 14, 15),
(4, 5, 6, 7, 12, 13, 14, 15),
(8, 9, 10, 11, 12, 13, 14, 15)]
[(0, 5, 6, 7, 8, 9, 14, 15),
(1, 4, 5, 6, 10, 13, 14, 15),
(1, 2, 6, 7, 8, 10, 11, 15),
(8, 9, 10, 11, 12, 13, 14, 15),
(0, 3, 4, 5, 9, 12, 13, 14),
(0, 2, 3, 7, 8, 9, 11, 12),
(1, 2, 3, 4, 10, 11, 12, 13),
(0, 1, 2, 3, 4, 5, 6, 7)]
sage: face = p.faces(3)[3]
sage: face.ambient_Hrepresentation()
(An inequality (1, 0, 0, 0) x + 1 >= 0,)
sage: face.vertices()
(A vertex at (-1, -1, -1, -1), A vertex at (-1, -1, -1, 1),
A vertex at (-1, -1, 1, -1), A vertex at (-1, -1, 1, 1),
A vertex at (-1, 1, -1, -1), A vertex at (-1, 1, -1, 1),
A vertex at (-1, 1, 1, -1), A vertex at (-1, 1, 1, 1))
(A vertex at (-1, -1, 1, -1),
A vertex at (-1, -1, 1, 1),
A vertex at (-1, 1, -1, -1),
A vertex at (-1, 1, 1, -1),
A vertex at (-1, 1, 1, 1),
A vertex at (-1, 1, -1, 1),
A vertex at (-1, -1, -1, 1),
A vertex at (-1, -1, -1, -1))
You can use the
:meth:`~sage.geometry.polyhedron.representation.PolyhedronRepresentation.index`
Expand All @@ -5866,19 +5872,19 @@ def faces(self, face_dimension):
sage: [get_idx(_) for _ in face.ambient_Hrepresentation()]
[4]
sage: [get_idx(_) for _ in face.ambient_Vrepresentation()]
[0, 1, 2, 3, 4, 5, 6, 7]
[8, 9, 10, 11, 12, 13, 14, 15]
sage: [ ([get_idx(_) for _ in face.ambient_Vrepresentation()],
....: [get_idx(_) for _ in face.ambient_Hrepresentation()])
....: for face in p.faces(3) ]
[([0, 2, 4, 6, 8, 10, 12, 14], [7]),
([0, 1, 4, 5, 8, 9, 12, 13], [6]),
([0, 1, 2, 3, 8, 9, 10, 11], [5]),
([0, 1, 2, 3, 4, 5, 6, 7], [4]),
([1, 3, 5, 7, 9, 11, 13, 15], [3]),
([2, 3, 6, 7, 10, 11, 14, 15], [2]),
([4, 5, 6, 7, 12, 13, 14, 15], [1]),
([8, 9, 10, 11, 12, 13, 14, 15], [0])]
[([0, 5, 6, 7, 8, 9, 14, 15], [7]),
([1, 4, 5, 6, 10, 13, 14, 15], [6]),
([1, 2, 6, 7, 8, 10, 11, 15], [5]),
([8, 9, 10, 11, 12, 13, 14, 15], [4]),
([0, 3, 4, 5, 9, 12, 13, 14], [3]),
([0, 2, 3, 7, 8, 9, 11, 12], [2]),
([1, 2, 3, 4, 10, 11, 12, 13], [1]),
([0, 1, 2, 3, 4, 5, 6, 7], [0])]
TESTS::
Expand Down Expand Up @@ -6498,7 +6504,7 @@ def one_point_suspension(self, vertex):
It works with a polyhedral face as well::
sage: vv = cube.faces(0)[0]
sage: vv = cube.faces(0)[1]
sage: ops_cube2 = cube.one_point_suspension(vv)
sage: ops_cube == ops_cube2
True
Expand Down Expand Up @@ -6674,7 +6680,7 @@ def schlegel_projection(self, projection_dir=None, height=1.1):
sage: schlegel_edge_indices = sch_proj.lines
sage: schlegel_edges = [sch_proj.coordinates_of(x) for x in schlegel_edge_indices]
sage: len([x for x in schlegel_edges if x[0][0] > 0])
4
5
"""
proj = self.projection()
if projection_dir is None:
Expand Down
72 changes: 36 additions & 36 deletions src/sage/geometry/polyhedron/combinatorial_polyhedron/base.pyx
View file
Original file line number Diff line number Diff line change
Expand Up @@ -35,16 +35,16 @@ Construction::
Obtaining edges and ridges::
sage: C.edges()[:2]
((A vertex at (1, 1, 1, -1), A vertex at (1, 1, 1, 1)),
(A vertex at (1, 1, -1, 1), A vertex at (1, 1, 1, 1)))
((A vertex at (-1, -1, -1, 1), A vertex at (-1, -1, -1, -1)),
(A vertex at (-1, 1, -1, -1), A vertex at (-1, -1, -1, -1)))
sage: C.edges(names=False)[:2]
((14, 15), (13, 15))
((14, 15), (10, 15))
sage: C.ridges()[:2]
((An inequality (0, 0, 1, 0) x + 1 >= 0,
An inequality (0, 0, 0, 1) x + 1 >= 0),
(An inequality (0, 1, 0, 0) x + 1 >= 0,
An inequality (0, 0, 0, 1) x + 1 >= 0))
An inequality (0, 1, 0, 0) x + 1 >= 0),
(An inequality (0, 0, 0, 1) x + 1 >= 0,
An inequality (0, 1, 0, 0) x + 1 >= 0))
sage: C.ridges(names=False)[:2]
((6, 7), (5, 7))
Expand Down Expand Up @@ -817,37 +817,37 @@ cdef class CombinatorialPolyhedron(SageObject):
sage: P = polytopes.cube()
sage: C = CombinatorialPolyhedron(P)
sage: C.facets()
((A vertex at (-1, -1, 1),
A vertex at (-1, 1, 1),
A vertex at (1, -1, 1),
A vertex at (1, 1, 1)),
(A vertex at (-1, 1, -1),
A vertex at (-1, 1, 1),
((A vertex at (1, -1, -1),
A vertex at (1, 1, -1),
A vertex at (1, 1, 1)),
(A vertex at (1, -1, -1),
A vertex at (1, 1, 1),
A vertex at (1, -1, 1)),
(A vertex at (1, 1, -1),
A vertex at (1, 1, 1),
A vertex at (-1, 1, -1),
A vertex at (-1, 1, 1)),
(A vertex at (1, 1, 1),
A vertex at (1, -1, 1),
A vertex at (1, 1, -1),
A vertex at (1, 1, 1)),
(A vertex at (-1, -1, -1),
A vertex at (-1, -1, 1),
A vertex at (-1, 1, -1),
A vertex at (-1, 1, 1)),
(A vertex at (-1, -1, -1),
(A vertex at (-1, -1, 1),
A vertex at (-1, -1, -1),
A vertex at (-1, 1, -1),
A vertex at (1, -1, -1),
A vertex at (1, 1, -1)),
(A vertex at (-1, -1, -1),
A vertex at (-1, 1, 1)),
(A vertex at (1, -1, -1),
A vertex at (1, 1, -1),
A vertex at (-1, -1, -1),
A vertex at (-1, 1, -1)),
(A vertex at (1, -1, -1),
A vertex at (1, -1, 1),
A vertex at (-1, -1, 1),
A vertex at (1, -1, -1),
A vertex at (1, -1, 1)))
A vertex at (-1, -1, -1)))
sage: C.facets(names=False)
((1, 3, 5, 7),
(2, 3, 6, 7),
((0, 1, 2, 3),
(1, 2, 6, 7),
(2, 3, 4, 7),
(4, 5, 6, 7),
(0, 1, 2, 3),
(0, 2, 4, 6),
(0, 1, 4, 5))
(0, 1, 5, 6),
(0, 3, 4, 5))
"""
if unlikely(self.dimension() == 0):
# Special attention for this trivial case.
Expand Down Expand Up @@ -889,14 +889,14 @@ cdef class CombinatorialPolyhedron(SageObject):
sage: P = polytopes.cube()
sage: C = P.combinatorial_polyhedron()
sage: C.incidence_matrix()
[1 0 0 0 1 1]
[1 1 0 0 1 0]
[1 1 1 0 0 0]
[1 0 1 0 0 1]
[0 0 1 1 0 1]
[0 0 0 1 1 1]
[1 0 0 1 0 1]
[0 1 0 1 1 0]
[1 1 0 1 0 0]
[0 0 1 0 1 1]
[1 0 1 0 0 1]
[0 1 1 0 1 0]
[1 1 1 0 0 0]
[0 1 1 1 0 0]
sage: P.incidence_matrix() == C.incidence_matrix()
True
Expand Down Expand Up @@ -2010,7 +2010,7 @@ cdef class CombinatorialPolyhedron(SageObject):
A 2-dimensional face of a 4-dimensional combinatorial polyhedron,
A 3-dimensional face of a 4-dimensional combinatorial polyhedron]
sage: [face.ambient_V_indices() for face in chain]
[(0,), (0, 8), (0, 4, 8, 12), (0, 2, 4, 6, 8, 10, 12, 14)]
[(15,), (6, 15), (5, 6, 14, 15), (0, 5, 6, 7, 8, 9, 14, 15)]
sage: P = polytopes.permutahedron(4)
sage: C = P.combinatorial_polyhedron()
Expand Down
32 changes: 16 additions & 16 deletions src/sage/geometry/polyhedron/combinatorial_polyhedron/face_iterator.pyx
View file
Original file line number Diff line number Diff line change
Expand Up @@ -258,13 +258,13 @@ cdef class FaceIterator(SageObject):
(1,),
(0,),
(3, 4),
(2, 4),
(1, 4),
(0, 4),
(1, 3, 4),
(1, 2, 4),
(0, 1, 4),
(2, 3),
(1, 3),
(0, 3),
(0, 1, 3),
(1, 2, 3),
(1, 2),
(0, 2),
(0, 1, 2),
Expand Down Expand Up @@ -294,22 +294,22 @@ cdef class FaceIterator(SageObject):
(1,),
(0,),
(6, 7),
(5, 7),
(3, 7),
(4, 7),
(2, 7),
(4, 5, 6, 7),
(2, 3, 6, 7),
(1, 3, 5, 7),
(4, 6),
(2, 6),
(0, 2, 4, 6),
(1, 2, 6, 7),
(2, 3, 4, 7),
(5, 6),
(1, 6),
(0, 1, 5, 6),
(4, 5),
(1, 5),
(0, 1, 4, 5),
(0, 4),
(0, 5),
(0, 3, 4, 5),
(3, 4),
(2, 3),
(1, 3),
(0, 3),
(0, 1, 2, 3),
(0, 2),
(1, 2),
(0, 1)]
sage: it = C.face_iter(dual=False)
Expand Down
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