def n_integral_points(self):
"""
Return the number of integral points.
OUTPUT:
Integer. The number of integral points contained in the
lattice polytope.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((0,0),(1,0),(0,1)).n_integral_points()
3
"""
if self.is_empty():
return tuple()
box_min, box_max = self.bounding_box()
from sage.geometry.integral_points import rectangular_box_points
return rectangular_box_points(list(box_min), list(box_max), self, count_only=True)
def n_integral_points(self):
"""
Return the number of integral points.
OUTPUT:
Integer. The number of integral points contained in the
lattice polytope.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((0,0),(1,0),(0,1)).n_integral_points()
3
"""
if self.is_empty():
return tuple()
box_min, box_max = self.bounding_box()
from sage.geometry.integral_points import rectangular_box_points
return rectangular_box_points(list(box_min), list(box_max), self, count_only=True)
def _integral_points_saturating(self):
"""
Return the integral points together with information about
which inequalities are saturated.
See :func:`~sage.geometry.integral_points.rectangular_box_points`.
OUTPUT:
A tuple of pairs (one for each integral point) consisting of a
pair ``(point, Hrep)``, where ``point`` is the coordinate
vector of the integral point and ``Hrep`` is the set of
indices of the :meth:`minimized_constraints` that are
saturated at the point.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: quad = LatticePolytope_PPL((-1,-1),(0,1),(1,0),(1,1))
sage: quad._integral_points_saturating()
(((-1, -1), frozenset({0, 1})),
((0, 0), frozenset()),
((0, 1), frozenset({0, 3})),
((1, 0), frozenset({1, 2})),
((1, 1), frozenset({2, 3})))
"""
if self.is_empty():
return tuple()
box_min, box_max = self.bounding_box()
from sage.geometry.integral_points import rectangular_box_points
points = rectangular_box_points(list(box_min),
list(box_max),
self,
return_saturated=True)
if not self.n_integral_points.is_in_cache():
self.n_integral_points.set_cache(len(points))
if not self.integral_points.is_in_cache():
self.integral_points.set_cache(tuple(p[0] for p in points))
return points
def _integral_points_saturating(self):
"""
Return the integral points together with information about
which inequalities are saturated.
See :func:`~sage.geometry.integral_points.rectangular_box_points`.
OUTPUT:
A tuple of pairs (one for each integral point) consisting of a
pair ``(point, Hrep)``, where ``point`` is the coordinate
vector of the intgeral point and ``Hrep`` is the set of
indices of the :meth:`minimized_constraints` that are
saturated at the point.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: quad = LatticePolytope_PPL((-1,-1),(0,1),(1,0),(1,1))
sage: quad._integral_points_saturating()
(((-1, -1), frozenset({0, 1})),
((0, 0), frozenset()),
((0, 1), frozenset({0, 3})),
((1, 0), frozenset({1, 2})),
((1, 1), frozenset({2, 3})))
"""
if self.is_empty():
return tuple()
box_min, box_max = self.bounding_box()
from sage.geometry.integral_points import rectangular_box_points
points = rectangular_box_points(list(box_min), list(box_max), self,
return_saturated=True)
if not self.n_integral_points.is_in_cache():
self.n_integral_points.set_cache(len(points))
if not self.integral_points.is_in_cache():
self.integral_points.set_cache(tuple(p[0] for p in points))
return points
def integral_points(self):
r"""
Return the integral points in the polyhedron.
Uses the naive algorithm (iterate over a rectangular bounding
box).
OUTPUT:
The list of integral points in the polyhedron. If the
polyhedron is not compact, a ``ValueError`` is raised.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((-1,-1),(1,0),(1,1),(0,1)).integral_points()
((-1, -1), (0, 0), (0, 1), (1, 0), (1, 1))
sage: simplex = LatticePolytope_PPL((1,2,3), (2,3,7), (-2,-3,-11))
sage: simplex.integral_points()
((-2, -3, -11), (0, 0, -2), (1, 2, 3), (2, 3, 7))
The polyhedron need not be full-dimensional::
sage: simplex = LatticePolytope_PPL((1,2,3,5), (2,3,7,5), (-2,-3,-11,5))
sage: simplex.integral_points()
((-2, -3, -11, 5), (0, 0, -2, 5), (1, 2, 3, 5), (2, 3, 7, 5))
sage: point = LatticePolytope_PPL((2,3,7))
sage: point.integral_points()
((2, 3, 7),)
sage: empty = LatticePolytope_PPL()
sage: empty.integral_points()
()
Here is a simplex where the naive algorithm of running over
all points in a rectangular bounding box no longer works fast
enough::
sage: v = [(1,0,7,-1), (-2,-2,4,-3), (-1,-1,-1,4), (2,9,0,-5), (-2,-1,5,1)]
sage: simplex = LatticePolytope_PPL(v); simplex
A 4-dimensional lattice polytope in ZZ^4 with 5 vertices
sage: len(simplex.integral_points())
49
Finally, the 3-d reflexive polytope number 4078::
sage: v = [(1,0,0), (0,1,0), (0,0,1), (0,0,-1), (0,-2,1),
....: (-1,2,-1), (-1,2,-2), (-1,1,-2), (-1,-1,2), (-1,-3,2)]
sage: P = LatticePolytope_PPL(*v)
sage: pts1 = P.integral_points() # Sage's own code
sage: pts2 = LatticePolytope(v).points() # PALP
sage: for p in pts1: p.set_immutable()
sage: set(pts1) == set(pts2)
True
sage: timeit('Polyhedron(v).integral_points()') # random output
sage: timeit('LatticePolytope(v).points()') # random output
sage: timeit('LatticePolytope_PPL(*v).integral_points()') # random output
"""
if self.is_empty():
return tuple()
box_min, box_max = self.bounding_box()
from sage.geometry.integral_points import rectangular_box_points
points = rectangular_box_points(list(box_min), list(box_max), self)
if not self.n_integral_points.is_in_cache():
self.n_integral_points.set_cache(len(points))
return points
def integral_points(self, threshold=10000):
r"""
Return the integral points in the polyhedron.
Uses either the naive algorithm (iterate over a rectangular
bounding box) or triangulation + Smith form.
INPUT:
- ``threshold`` -- integer (default: 10000); use the naïve
algorithm as long as the bounding box is smaller than this
OUTPUT:
The list of integral points in the polyhedron. If the
polyhedron is not compact, a ``ValueError`` is raised.
EXAMPLES::
sage: Polyhedron(vertices=[(-1,-1), (1,0), (1,1), (0,1)], # optional - pynormaliz
....: backend='normaliz').integral_points()
((-1, -1), (0, 0), (0, 1), (1, 0), (1, 1))
sage: simplex = Polyhedron([(1,2,3), (2,3,7), (-2,-3,-11)], # optional - pynormaliz
....: backend='normaliz')
sage: simplex.integral_points() # optional - pynormaliz
((-2, -3, -11), (0, 0, -2), (1, 2, 3), (2, 3, 7))
The polyhedron need not be full-dimensional::
sage: simplex = Polyhedron([(1,2,3,5), (2,3,7,5), (-2,-3,-11,5)], # optional - pynormaliz
....: backend='normaliz')
sage: simplex.integral_points() # optional - pynormaliz
((-2, -3, -11, 5), (0, 0, -2, 5), (1, 2, 3, 5), (2, 3, 7, 5))
sage: point = Polyhedron([(2,3,7)], # optional - pynormaliz
....: backend='normaliz')
sage: point.integral_points() # optional - pynormaliz
((2, 3, 7),)
sage: empty = Polyhedron(backend='normaliz') # optional - pynormaliz
sage: empty.integral_points() # optional - pynormaliz
()
Here is a simplex where the naive algorithm of running over
all points in a rectangular bounding box no longer works fast
enough::
sage: v = [(1,0,7,-1), (-2,-2,4,-3), (-1,-1,-1,4), (2,9,0,-5), (-2,-1,5,1)]
sage: simplex = Polyhedron(v, backend='normaliz'); simplex # optional - pynormaliz
A 4-dimensional polyhedron in ZZ^4 defined as the convex hull of 5 vertices
sage: len(simplex.integral_points()) # optional - pynormaliz
49
A rather thin polytope for which the bounding box method would
be a very bad idea (note this is a rational (non-lattice)
polytope, so the other backends use the bounding box method)::
sage: P = Polyhedron(vertices=((0, 0), (178933,37121))) + 1/1000*polytopes.hypercube(2)
sage: P = Polyhedron(vertices=P.vertices_list(), # optional - pynormaliz
....: backend='normaliz')
sage: len(P.integral_points()) # optional - pynormaliz
434
Finally, the 3-d reflexive polytope number 4078::
sage: v = [(1,0,0), (0,1,0), (0,0,1), (0,0,-1), (0,-2,1),
....: (-1,2,-1), (-1,2,-2), (-1,1,-2), (-1,-1,2), (-1,-3,2)]
sage: P = Polyhedron(v, backend='normaliz') # optional - pynormaliz
sage: pts1 = P.integral_points() # optional - pynormaliz
sage: all(P.contains(p) for p in pts1) # optional - pynormaliz
True
sage: pts2 = LatticePolytope(v).points() # PALP
sage: for p in pts1: p.set_immutable() # optional - pynormaliz
sage: set(pts1) == set(pts2) # optional - pynormaliz
True
sage: timeit('Polyhedron(v, backend='normaliz').integral_points()') # not tested - random
625 loops, best of 3: 1.41 ms per loop
sage: timeit('LatticePolytope(v).points()') # not tested - random
25 loops, best of 3: 17.2 ms per loop
TESTS:
Test some trivial cases (see :trac:`17937`):
Empty polyhedron in 1 dimension::
sage: P = Polyhedron(ambient_dim=1, backend='normaliz') # optional - pynormaliz
sage: P.integral_points() # optional - pynormaliz
()
Empty polyhedron in 0 dimensions::
sage: P = Polyhedron(ambient_dim=0, backend='normaliz') # optional - pynormaliz
sage: P.integral_points() # optional - pynormaliz
()
Single point in 1 dimension::
sage: P = Polyhedron([[3]], backend='normaliz') # optional - pynormaliz
sage: P.integral_points() # optional - pynormaliz
((3),)
Single non-integral point in 1 dimension::
sage: P = Polyhedron([[1/2]], backend='normaliz') # optional - pynormaliz
sage: P.integral_points() # optional - pynormaliz
()
Single point in 0 dimensions::
sage: P = Polyhedron([[]], backend='normaliz') # optional - pynormaliz
sage: P.integral_points() # optional - pynormaliz
((),)
A polytope with no integral points (:trac:`22938`)::
sage: ieqs = [[1, 2, -1, 0], [0, -1, 2, -1], [0, 0, -1, 2],
....: [0, -1, 0, 0], [0, 0, -1, 0], [0, 0, 0, -1],
....: [-1, -1, -1, -1], [1, 1, 0, 0], [1, 0, 1, 0],
....: [1, 0, 0, 1]]
sage: P = Polyhedron(ieqs=ieqs, backend='normaliz') # optional - pynormaliz
sage: P.bounding_box() # optional - pynormaliz
((-3/4, -1/2, -1/4), (-1/2, -1/4, 0))
sage: P.bounding_box(integral_hull=True) # optional - pynormaliz
(None, None)
sage: P.integral_points() # optional - pynormaliz
()
"""
import PyNormaliz
if not self.is_compact():
raise ValueError(
'can only enumerate points in a compact polyhedron')
# Trivial cases: polyhedron with 0 or 1 vertices
if self.n_vertices() == 0:
return ()
if self.n_vertices() == 1:
v = self.vertices_list()[0]
try:
return (vector(ZZ, v), )
except TypeError: # vertex not integral
return ()
# for small bounding boxes, it is faster to naively iterate over the points of the box
if threshold > 1:
box_min, box_max = self.bounding_box(integral_hull=True)
if box_min is None:
return ()
box_points = prod(
max_coord - min_coord + 1
for min_coord, max_coord in zip(box_min, box_max))
if box_points < threshold:
from sage.geometry.integral_points import rectangular_box_points
return rectangular_box_points(list(box_min), list(box_max),
self)
# Compute with normaliz
points = []
cone = self._normaliz_cone
assert cone
for g in PyNormaliz.NmzResult(cone, "ModuleGenerators"):
assert g[-1] == 1
points.append(vector(ZZ, g[:-1]))
return tuple(points)
def integral_points(self, threshold=10000):
r"""
Return the integral points in the polyhedron.
Uses either the naive algorithm (iterate over a rectangular
bounding box) or triangulation + Smith form.
INPUT:
- ``threshold`` -- integer (default: 10000); use the naïve
algorithm as long as the bounding box is smaller than this
OUTPUT:
The list of integral points in the polyhedron. If the
polyhedron is not compact, a ``ValueError`` is raised.
EXAMPLES::
sage: Polyhedron(vertices=[(-1,-1), (1,0), (1,1), (0,1)], # optional - pynormaliz
....: backend='normaliz').integral_points()
((-1, -1), (0, 0), (0, 1), (1, 0), (1, 1))
sage: simplex = Polyhedron([(1,2,3), (2,3,7), (-2,-3,-11)], # optional - pynormaliz
....: backend='normaliz')
sage: simplex.integral_points() # optional - pynormaliz
((-2, -3, -11), (0, 0, -2), (1, 2, 3), (2, 3, 7))
The polyhedron need not be full-dimensional::
sage: simplex = Polyhedron([(1,2,3,5), (2,3,7,5), (-2,-3,-11,5)], # optional - pynormaliz
....: backend='normaliz')
sage: simplex.integral_points() # optional - pynormaliz
((-2, -3, -11, 5), (0, 0, -2, 5), (1, 2, 3, 5), (2, 3, 7, 5))
sage: point = Polyhedron([(2,3,7)], # optional - pynormaliz
....: backend='normaliz')
sage: point.integral_points() # optional - pynormaliz
((2, 3, 7),)
sage: empty = Polyhedron(backend='normaliz') # optional - pynormaliz
sage: empty.integral_points() # optional - pynormaliz
()
Here is a simplex where the naive algorithm of running over
all points in a rectangular bounding box no longer works fast
enough::
sage: v = [(1,0,7,-1), (-2,-2,4,-3), (-1,-1,-1,4), (2,9,0,-5), (-2,-1,5,1)]
sage: simplex = Polyhedron(v, backend='normaliz'); simplex # optional - pynormaliz
A 4-dimensional polyhedron in ZZ^4 defined as the convex hull of 5 vertices
sage: len(simplex.integral_points()) # optional - pynormaliz
49
A rather thin polytope for which the bounding box method would
be a very bad idea (note this is a rational (non-lattice)
polytope, so the other backends use the bounding box method)::
sage: P = Polyhedron(vertices=((0, 0), (178933,37121))) + 1/1000*polytopes.hypercube(2)
sage: P = Polyhedron(vertices=P.vertices_list(), # optional - pynormaliz
....: backend='normaliz')
sage: len(P.integral_points()) # optional - pynormaliz
434
Finally, the 3-d reflexive polytope number 4078::
sage: v = [(1,0,0), (0,1,0), (0,0,1), (0,0,-1), (0,-2,1),
....: (-1,2,-1), (-1,2,-2), (-1,1,-2), (-1,-1,2), (-1,-3,2)]
sage: P = Polyhedron(v, backend='normaliz') # optional - pynormaliz
sage: pts1 = P.integral_points() # optional - pynormaliz
sage: all(P.contains(p) for p in pts1) # optional - pynormaliz
True
sage: pts2 = LatticePolytope(v).points() # PALP
sage: for p in pts1: p.set_immutable() # optional - pynormaliz
sage: set(pts1) == set(pts2) # optional - pynormaliz
True
sage: timeit('Polyhedron(v, backend='normaliz').integral_points()') # not tested - random
625 loops, best of 3: 1.41 ms per loop
sage: timeit('LatticePolytope(v).points()') # not tested - random
25 loops, best of 3: 17.2 ms per loop
TESTS:
Test some trivial cases (see :trac:`17937`):
Empty polyhedron in 1 dimension::
sage: P = Polyhedron(ambient_dim=1, backend='normaliz') # optional - pynormaliz
sage: P.integral_points() # optional - pynormaliz
()
Empty polyhedron in 0 dimensions::
sage: P = Polyhedron(ambient_dim=0, backend='normaliz') # optional - pynormaliz
sage: P.integral_points() # optional - pynormaliz
()
Single point in 1 dimension::
sage: P = Polyhedron([[3]], backend='normaliz') # optional - pynormaliz
sage: P.integral_points() # optional - pynormaliz
((3),)
Single non-integral point in 1 dimension::
sage: P = Polyhedron([[1/2]], backend='normaliz') # optional - pynormaliz
sage: P.integral_points() # optional - pynormaliz
()
Single point in 0 dimensions::
sage: P = Polyhedron([[]], backend='normaliz') # optional - pynormaliz
sage: P.integral_points() # optional - pynormaliz
((),)
A polytope with no integral points (:trac:`22938`)::
sage: ieqs = [[1, 2, -1, 0], [0, -1, 2, -1], [0, 0, -1, 2],
....: [0, -1, 0, 0], [0, 0, -1, 0], [0, 0, 0, -1],
....: [-1, -1, -1, -1], [1, 1, 0, 0], [1, 0, 1, 0],
....: [1, 0, 0, 1]]
sage: P = Polyhedron(ieqs=ieqs, backend='normaliz') # optional - pynormaliz
sage: P.bounding_box() # optional - pynormaliz
((-3/4, -1/2, -1/4), (-1/2, -1/4, 0))
sage: P.bounding_box(integral_hull=True) # optional - pynormaliz
(None, None)
sage: P.integral_points() # optional - pynormaliz
()
Check the polytopes from :trac:`22984`::
sage: base = [[0, 2, 0, -1, 0, 0, 0, 0, 0],
....: [0, 0, 2, 0, -1, 0, 0, 0, 0],
....: [1, -1, 0, 2, -1, 0, 0, 0, 0],
....: [0, 0, -1, -1, 2, -1, 0, 0, 0],
....: [0, 0, 0, 0, -1, 2, -1, 0, 0],
....: [0, 0, 0, 0, 0, -1, 2, -1, 0],
....: [1, 0, 0, 0, 0, 0, -1, 2, -1],
....: [0, 0, 0, 0, 0, 0, 0, -1, 2],
....: [0, -1, 0, 0, 0, 0, 0, 0, 0],
....: [0, 0, -1, 0, 0, 0, 0, 0, 0],
....: [0, 0, 0, -1, 0, 0, 0, 0, 0],
....: [0, 0, 0, 0, -1, 0, 0, 0, 0],
....: [0, 0, 0, 0, 0, -1, 0, 0, 0],
....: [0, 0, 0, 0, 0, 0, -1, 0, 0],
....: [0, 0, 0, 0, 0, 0, 0, -1, 0],
....: [0, 0, 0, 0, 0, 0, 0, 0, -1],
....: [-1, -1, -1, -1, -1, -1, -1, -1, -1]]
sage: ieqs = base + [
....: [2, 1, 0, 0, 0, 0, 0, 0, 0],
....: [4, 0, 1, 0, 0, 0, 0, 0, 0],
....: [4, 0, 0, 1, 0, 0, 0, 0, 0],
....: [7, 0, 0, 0, 1, 0, 0, 0, 0],
....: [6, 0, 0, 0, 0, 1, 0, 0, 0],
....: [4, 0, 0, 0, 0, 0, 1, 0, 0],
....: [2, 0, 0, 0, 0, 0, 0, 1, 0],
....: [1, 0, 0, 0, 0, 0, 0, 0, 1]]
sage: P = Polyhedron(ieqs=ieqs, backend='normaliz') # optional - pynormaliz
sage: P.integral_points() # optional - pynormaliz
((-2, -2, -4, -5, -4, -3, -2, -1),
(-2, -2, -4, -5, -4, -3, -2, 0),
(-1, -2, -3, -4, -3, -2, -2, -1),
(-1, -2, -3, -4, -3, -2, -1, 0),
(-1, -1, -2, -2, -2, -2, -2, -1),
(-1, -1, -2, -2, -1, -1, -1, 0),
(-1, -1, -2, -2, -1, 0, 0, 0),
(-1, 0, -2, -2, -2, -2, -2, -1),
(0, -1, -1, -2, -2, -2, -2, -1),
(0, 0, -1, -1, -1, -1, -1, 0))
sage: ieqs = base + [
....: [3, 1, 0, 0, 0, 0, 0, 0, 0],
....: [4, 0, 1, 0, 0, 0, 0, 0, 0],
....: [6, 0, 0, 1, 0, 0, 0, 0, 0],
....: [8, 0, 0, 0, 1, 0, 0, 0, 0],
....: [6, 0, 0, 0, 0, 1, 0, 0, 0],
....: [4, 0, 0, 0, 0, 0, 1, 0, 0],
....: [2, 0, 0, 0, 0, 0, 0, 1, 0],
....: [1, 0, 0, 0, 0, 0, 0, 0, 1]]
sage: P = Polyhedron(ieqs=ieqs, backend='normaliz') # optional - pynormaliz
sage: P.integral_points() # optional - pynormaliz
((-3, -4, -6, -8, -6, -4, -2, -1),
(-3, -4, -6, -8, -6, -4, -2, 0),
(-2, -2, -4, -5, -4, -3, -2, -1),
(-2, -2, -4, -5, -4, -3, -2, 0),
(-1, -2, -3, -4, -3, -2, -2, -1),
(-1, -2, -3, -4, -3, -2, -1, 0),
(-1, -1, -2, -2, -2, -2, -2, -1),
(-1, -1, -2, -2, -1, -1, -1, 0),
(-1, -1, -2, -2, -1, 0, 0, 0),
(-1, 0, -2, -2, -2, -2, -2, -1),
(0, -1, -1, -2, -2, -2, -2, -1),
(0, 0, -1, -1, -1, -1, -1, 0))
"""
import PyNormaliz
if not self.is_compact():
raise ValueError('can only enumerate points in a compact polyhedron')
# Trivial cases: polyhedron with 0 or 1 vertices
if self.n_vertices() == 0:
return ()
if self.n_vertices() == 1:
v = self.vertices_list()[0]
try:
return (vector(ZZ, v),)
except TypeError: # vertex not integral
return ()
# for small bounding boxes, it is faster to naively iterate over the points of the box
if threshold > 1:
box_min, box_max = self.bounding_box(integral_hull=True)
if box_min is None:
return ()
box_points = prod(max_coord-min_coord+1 for min_coord, max_coord in zip(box_min, box_max))
if box_points<threshold:
from sage.geometry.integral_points import rectangular_box_points
return rectangular_box_points(list(box_min), list(box_max), self)
# Compute with normaliz
points = []
cone = self._normaliz_cone
assert cone
for g in PyNormaliz.NmzResult(cone, "ModuleGenerators"):
assert g[-1] == 1
points.append(vector(ZZ, g[:-1]))
return tuple(points)
def integral_points(self):
r"""
Return the integral points in the polyhedron.
Uses the naive algorithm (iterate over a rectangular bounding
box).
OUTPUT:
The list of integral points in the polyhedron. If the
polyhedron is not compact, a ``ValueError`` is raised.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((-1,-1),(1,0),(1,1),(0,1)).integral_points()
((-1, -1), (0, 0), (0, 1), (1, 0), (1, 1))
sage: simplex = LatticePolytope_PPL((1,2,3), (2,3,7), (-2,-3,-11))
sage: simplex.integral_points()
((-2, -3, -11), (0, 0, -2), (1, 2, 3), (2, 3, 7))
The polyhedron need not be full-dimensional::
sage: simplex = LatticePolytope_PPL((1,2,3,5), (2,3,7,5), (-2,-3,-11,5))
sage: simplex.integral_points()
((-2, -3, -11, 5), (0, 0, -2, 5), (1, 2, 3, 5), (2, 3, 7, 5))
sage: point = LatticePolytope_PPL((2,3,7))
sage: point.integral_points()
((2, 3, 7),)
sage: empty = LatticePolytope_PPL()
sage: empty.integral_points()
()
Here is a simplex where the naive algorithm of running over
all points in a rectangular bounding box no longer works fast
enough::
sage: v = [(1,0,7,-1), (-2,-2,4,-3), (-1,-1,-1,4), (2,9,0,-5), (-2,-1,5,1)]
sage: simplex = LatticePolytope_PPL(v); simplex
A 4-dimensional lattice polytope in ZZ^4 with 5 vertices
sage: len(simplex.integral_points())
49
Finally, the 3-d reflexive polytope number 4078::
sage: v = [(1,0,0), (0,1,0), (0,0,1), (0,0,-1), (0,-2,1),
....: (-1,2,-1), (-1,2,-2), (-1,1,-2), (-1,-1,2), (-1,-3,2)]
sage: P = LatticePolytope_PPL(*v)
sage: pts1 = P.integral_points() # Sage's own code
sage: pts2 = LatticePolytope(v).points() # PALP
sage: for p in pts1: p.set_immutable()
sage: set(pts1) == set(pts2)
True
sage: timeit('Polyhedron(v).integral_points()') # random output
sage: timeit('LatticePolytope(v).points()') # random output
sage: timeit('LatticePolytope_PPL(*v).integral_points()') # random output
"""
if self.is_empty():
return tuple()
box_min, box_max = self.bounding_box()
from sage.geometry.integral_points import rectangular_box_points
points = rectangular_box_points(list(box_min), list(box_max), self)
if not self.n_integral_points.is_in_cache():
self.n_integral_points.set_cache(len(points))
return points
def integral_points(self, threshold=100000):
r"""
Return the integral points in the polyhedron.
Uses either the naive algorithm (iterate over a rectangular
bounding box) or triangulation + Smith form.
INPUT:
- ``threshold`` -- integer (default: 100000). Use the naive
algorithm as long as the bounding box is smaller than this.
OUTPUT:
The list of integral points in the polyhedron. If the
polyhedron is not compact, a ``ValueError`` is raised.
EXAMPLES::
sage: Polyhedron(vertices=[(-1,-1),(1,0),(1,1),(0,1)]).integral_points()
((-1, -1), (0, 0), (0, 1), (1, 0), (1, 1))
sage: simplex = Polyhedron([(1,2,3), (2,3,7), (-2,-3,-11)])
sage: simplex.integral_points()
((-2, -3, -11), (0, 0, -2), (1, 2, 3), (2, 3, 7))
The polyhedron need not be full-dimensional::
sage: simplex = Polyhedron([(1,2,3,5), (2,3,7,5), (-2,-3,-11,5)])
sage: simplex.integral_points()
((-2, -3, -11, 5), (0, 0, -2, 5), (1, 2, 3, 5), (2, 3, 7, 5))
sage: point = Polyhedron([(2,3,7)])
sage: point.integral_points()
((2, 3, 7),)
sage: empty = Polyhedron()
sage: empty.integral_points()
()
Here is a simplex where the naive algorithm of running over
all points in a rectangular bounding box no longer works fast
enough::
sage: v = [(1,0,7,-1), (-2,-2,4,-3), (-1,-1,-1,4), (2,9,0,-5), (-2,-1,5,1)]
sage: simplex = Polyhedron(v); simplex
A 4-dimensional polyhedron in ZZ^4 defined as the convex hull of 5 vertices
sage: len(simplex.integral_points())
49
A case where rounding in the right direction goes a long way::
sage: P = 1/10*polytopes.hypercube(14, backend='field')
sage: P.integral_points()
((0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),)
Finally, the 3-d reflexive polytope number 4078::
sage: v = [(1,0,0), (0,1,0), (0,0,1), (0,0,-1), (0,-2,1),
....: (-1,2,-1), (-1,2,-2), (-1,1,-2), (-1,-1,2), (-1,-3,2)]
sage: P = Polyhedron(v)
sage: pts1 = P.integral_points() # Sage's own code
sage: all(P.contains(p) for p in pts1)
True
sage: pts2 = LatticePolytope(v).points() # PALP
sage: for p in pts1: p.set_immutable()
sage: set(pts1) == set(pts2)
True
sage: timeit('Polyhedron(v).integral_points()') # not tested - random
625 loops, best of 3: 1.41 ms per loop
sage: timeit('LatticePolytope(v).points()') # not tested - random
25 loops, best of 3: 17.2 ms per loop
TESTS:
Test some trivial cases (see :trac:`17937`)::
sage: P = Polyhedron(ambient_dim=1) # empty polyhedron in 1 dimension
sage: P.integral_points()
()
sage: P = Polyhedron(ambient_dim=0) # empty polyhedron in 0 dimensions
sage: P.integral_points()
()
sage: P = Polyhedron([[3]]) # single point in 1 dimension
sage: P.integral_points()
((3),)
sage: P = Polyhedron([[1/2]]) # single non-integral point in 1 dimension
sage: P.integral_points()
()
sage: P = Polyhedron([[]]) # single point in 0 dimensions
sage: P.integral_points()
((),)
Test unbounded polyhedron::
sage: P = Polyhedron(rays=[[1,0,0]])
sage: P.integral_points()
Traceback (most recent call last):
...
ValueError: can only enumerate points in a compact polyhedron
"""
from sage.misc.misc_c import prod
if not self.is_compact():
raise ValueError('can only enumerate points in a compact polyhedron')
# Trivial cases: polyhedron with 0 or 1 vertices
if self.n_vertices() == 0:
return ()
if self.n_vertices() == 1:
v = self.vertices_list()[0]
try:
return (vector(ZZ, v),)
except TypeError: # vertex not integral
return ()
# for small bounding boxes, it is faster to naively iterate over the points of the box
box_min, box_max = self.bounding_box(integral_hull=True)
if box_min is None:
return ()
box_points = prod(max_coord-min_coord+1 for min_coord, max_coord in zip(box_min, box_max))
if not self.is_lattice_polytope() or \
(self.is_simplex() and box_points < 1000) or \
box_points < threshold:
from sage.geometry.integral_points import rectangular_box_points
return rectangular_box_points(list(box_min), list(box_max), self)
# for more complicate polytopes, triangulate & use smith normal form
from sage.geometry.integral_points import simplex_points
if self.is_simplex():
return simplex_points(self.Vrepresentation())
triangulation = self.triangulate()
points = set()
for simplex in triangulation:
triang_vertices = [self.Vrepresentation(i) for i in simplex]
new_points = simplex_points(triang_vertices)
for p in new_points:
p.set_immutable()
points.update(new_points)
# assert all(self.contains(p) for p in points) # slow
return tuple(points)