Пример #1
0
    def _init_from_Vrepresentation(self,
                                   ambient_dim,
                                   vertices,
                                   rays,
                                   lines,
                                   minimize=True):
        """
        Construct polyhedron from V-representation data.

        INPUT:

        - ``ambient_dim`` -- integer. The dimension of the ambient space.
        
        - ``vertices`` -- list of point. Each point can be specified
           as any iterable container of
           :meth:`~sage.geometry.polyhedron.base.base_ring` elements.
        
        - ``rays`` -- list of rays. Each ray can be specified as any
          iterable container of
          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.
        
        - ``lines`` -- list of lines. Each line can be specified as
          any iterable container of
          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.

        EXAMPLES::

            sage: p = Polyhedron(backend='ppl')
            sage: from sage.geometry.polyhedron.backend_ppl import Polyhedron_QQ_ppl
            sage: Polyhedron_QQ_ppl._init_from_Vrepresentation(p, 2, [], [], [])
        """
        gs = Generator_System()
        if vertices is None: vertices = []
        for v in vertices:
            d = lcm([denominator(v_i) for v_i in v])
            dv = [ZZ(d * v_i) for v_i in v]
            gs.insert(point(Linear_Expression(dv, 0), d))
        if rays is None: rays = []
        for r in rays:
            d = lcm([denominator(r_i) for r_i in r])
            dr = [ZZ(d * r_i) for r_i in r]
            gs.insert(ray(Linear_Expression(dr, 0)))
        if lines is None: lines = []
        for l in lines:
            d = lcm([denominator(l_i) for l_i in l])
            dl = [ZZ(d * l_i) for l_i in l]
            gs.insert(line(Linear_Expression(dl, 0)))
        self._ppl_polyhedron = C_Polyhedron(gs)
        self._init_Vrepresentation_from_ppl(minimize)
        self._init_Hrepresentation_from_ppl(minimize)
Пример #2
0
    def _init_from_Vrepresentation(self, ambient_dim, vertices, rays, lines, minimize=True):
        """
        Construct polyhedron from V-representation data.

        INPUT:

        - ``ambient_dim`` -- integer. The dimension of the ambient space.
        
        - ``vertices`` -- list of point. Each point can be specified
           as any iterable container of
           :meth:`~sage.geometry.polyhedron.base.base_ring` elements.
        
        - ``rays`` -- list of rays. Each ray can be specified as any
          iterable container of
          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.
        
        - ``lines`` -- list of lines. Each line can be specified as
          any iterable container of
          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.

        EXAMPLES::

            sage: p = Polyhedron(backend='ppl')
            sage: from sage.geometry.polyhedron.backend_ppl import Polyhedron_QQ_ppl
            sage: Polyhedron_QQ_ppl._init_from_Vrepresentation(p, 2, [], [], [])
        """
        gs = Generator_System()
        if vertices is None: vertices = []
        for v in vertices:
            d = lcm([denominator(v_i) for v_i in v])
            dv = [ ZZ(d*v_i) for v_i in v ]
            gs.insert(point(Linear_Expression(dv, 0), d))
        if rays is None: rays = []
        for r in rays:
            d = lcm([denominator(r_i) for r_i in r])
            dr = [ ZZ(d*r_i) for r_i in r ]
            gs.insert(ray(Linear_Expression(dr, 0)))
        if lines is None: lines = []
        for l in lines:
            d = lcm([denominator(l_i) for l_i in l])
            dl = [ ZZ(d*l_i) for l_i in l ]
            gs.insert(line(Linear_Expression(dl, 0)))
        self._ppl_polyhedron = C_Polyhedron(gs)
        self._init_Vrepresentation_from_ppl(minimize)
        self._init_Hrepresentation_from_ppl(minimize)
Пример #3
0
    def fibration_generator(self, dim):
        """
        Generate the lattice polytope fibrations.

        For the purposes of this function, a lattice polytope fiber is
        a sub-lattice polytope. Projecting the plane spanned by the
        subpolytope to a point yields another lattice polytope, the
        base of the fibration.

        INPUT:

        - ``dim`` -- integer. The dimension of the lattice polytope
          fiber.

        OUTPUT:

        A generator yielding the distinct lattice polytope fibers of
        given dimension.

        EXAMPLES::

            sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
            sage: p = LatticePolytope_PPL((-9,-6,-1,-1),(0,0,0,1),(0,0,1,0),(0,1,0,0),(1,0,0,0))
            sage: list( p.fibration_generator(2) )
            [A 2-dimensional lattice polytope in ZZ^4 with 3 vertices]
        """
        assert self.is_full_dimensional()
        codim = self.space_dimension() - dim
        # "points" are the potential vertices of the fiber. They are
        # in the $codim$-skeleton of the polytope, which is contained
        # in the points that saturate at least $dim$ equations.
        points = [ p for p in self._integral_points_saturating() if len(p[1])>=dim ]
        points = sorted(points, key=lambda x:len(x[1]))

        # iterate over point combinations subject to all points being on one facet.
        def point_combinations_iterator(n, i0=0, saturated=None):
            for i in range(i0, len(points)):
                p, ieqs = points[i]
                if saturated is None:
                    saturated_ieqs = ieqs
                else:
                    saturated_ieqs = saturated.intersection(ieqs)
                if len(saturated_ieqs)==0:
                    continue
                if n == 1:
                    yield [i]
                else:
                    for c in point_combinations_iterator(n-1, i+1, saturated_ieqs):
                        yield [i] + c

        point_lines = [ line(Linear_Expression(p[0].list(),0)) for p in points ]
        origin = point()
        fibers = set()
        gs = Generator_System()
        for indices in point_combinations_iterator(dim):
            gs.clear()
            gs.insert(origin)
            for i in indices:
                gs.insert(point_lines[i])
            plane = C_Polyhedron(gs)
            if plane.affine_dimension() != dim:
                continue
            plane.intersection_assign(self)
            if (not self.is_full_dimensional()) and (plane.affine_dimension() != dim):
                continue
            try:
                fiber = LatticePolytope_PPL(plane)
            except TypeError:   # not a lattice polytope
                continue
            fiber_vertices = tuple(sorted(fiber.vertices()))
            if fiber_vertices not in fibers:
                yield fiber
                fibers.update([fiber_vertices])
Пример #4
0
def LatticePolytope_PPL(*args):
    """
    Construct a new instance of the PPL-based lattice polytope class.

    EXAMPLES::

        sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
        sage: LatticePolytope_PPL((0,0),(1,0),(0,1))
        A 2-dimensional lattice polytope in ZZ^2 with 3 vertices

        sage: from sage.libs.ppl import point, Generator_System, C_Polyhedron, Linear_Expression, Variable
        sage: p = point(Linear_Expression([2,3],0));  p
        point(2/1, 3/1)
        sage: LatticePolytope_PPL(p)
        A 0-dimensional lattice polytope in ZZ^2 with 1 vertex

        sage: P = C_Polyhedron(Generator_System(p));  P
        A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point
        sage: LatticePolytope_PPL(P)
        A 0-dimensional lattice polytope in ZZ^2 with 1 vertex

    A ``TypeError`` is raised if the arguments do not specify a lattice polytope::

        sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
        sage: LatticePolytope_PPL((0,0),(1/2,1))
        Traceback (most recent call last):
        ...
        TypeError: no conversion of this rational to integer

        sage: from sage.libs.ppl import point, Generator_System, C_Polyhedron, Linear_Expression, Variable
        sage: p = point(Linear_Expression([2,3],0), 5);  p
        point(2/5, 3/5)
        sage: LatticePolytope_PPL(p)
        Traceback (most recent call last):
         ...
        TypeError: generator is not a lattice polytope generator

        sage: P = C_Polyhedron(Generator_System(p));  P
        A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point
        sage: LatticePolytope_PPL(P)
        Traceback (most recent call last):
        ...
        TypeError: polyhedron has non-integral generators
    """
    polytope_class = LatticePolytope_PPL_class
    if len(args)==1 and isinstance(args[0], C_Polyhedron):
        polyhedron = args[0]
        polytope_class = _class_for_LatticePolytope(polyhedron.space_dimension())
        if not all(p.is_point() and p.divisor().is_one() for p in polyhedron.generators()):
            raise TypeError('polyhedron has non-integral generators')
        return polytope_class(polyhedron)
    if len(args)==1 \
            and isinstance(args[0], (list, tuple)) \
            and isinstance(args[0][0], (list,tuple)):
        vertices = args[0]
    else:
        vertices = args
    gs = Generator_System()
    for v in vertices:
        if isinstance(v, Generator):
            if (not v.is_point()) or (not v.divisor().is_one()):
                raise TypeError('generator is not a lattice polytope generator')
            gs.insert(v)
        else:
            gs.insert(point(Linear_Expression(v, 0)))
    if not gs.empty():
        dim = next(Generator_System_iterator(gs)).space_dimension()
        polytope_class = _class_for_LatticePolytope(dim)
    return polytope_class(gs)
Пример #5
0
    def _init_from_Vrepresentation(self,
                                   vertices,
                                   rays,
                                   lines,
                                   minimize=True,
                                   verbose=False):
        """
        Construct polyhedron from V-representation data.

        INPUT:

        - ``vertices`` -- list of point. Each point can be specified
           as any iterable container of
           :meth:`~sage.geometry.polyhedron.base.base_ring` elements.

        - ``rays`` -- list of rays. Each ray can be specified as any
          iterable container of
          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.

        - ``lines`` -- list of lines. Each line can be specified as
          any iterable container of
          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.

        - ``verbose`` -- boolean (default: ``False``). Whether to print
          verbose output for debugging purposes.

        EXAMPLES::

            sage: p = Polyhedron(backend='ppl')
            sage: from sage.geometry.polyhedron.backend_ppl import Polyhedron_ppl
            sage: Polyhedron_ppl._init_from_Vrepresentation(p, [], [], [])
        """
        gs = Generator_System()
        if vertices is None: vertices = []
        for v in vertices:
            d = LCM_list([denominator(v_i) for v_i in v])
            if d.is_one():
                gs.insert(point(Linear_Expression(v, 0)))
            else:
                dv = [d * v_i for v_i in v]
                gs.insert(point(Linear_Expression(dv, 0), d))
        if rays is None: rays = []
        for r in rays:
            d = LCM_list([denominator(r_i) for r_i in r])
            if d.is_one():
                gs.insert(ray(Linear_Expression(r, 0)))
            else:
                dr = [d * r_i for r_i in r]
                gs.insert(ray(Linear_Expression(dr, 0)))
        if lines is None: lines = []
        for l in lines:
            d = LCM_list([denominator(l_i) for l_i in l])
            if d.is_one():
                gs.insert(line(Linear_Expression(l, 0)))
            else:
                dl = [d * l_i for l_i in l]
                gs.insert(line(Linear_Expression(dl, 0)))
        if gs.empty():
            self._ppl_polyhedron = C_Polyhedron(self.ambient_dim(), 'empty')
        else:
            self._ppl_polyhedron = C_Polyhedron(gs)
        self._init_Vrepresentation_from_ppl(minimize)
        self._init_Hrepresentation_from_ppl(minimize)
Пример #6
0
    def fibration_generator(self, dim):
        """
        Generate the lattice polytope fibrations.

        For the purposes of this function, a lattice polytope fiber is
        a sub-lattice polytope. Projecting the plane spanned by the
        subpolytope to a point yields another lattice polytope, the
        base of the fibration.

        INPUT:

        - ``dim`` -- integer. The dimension of the lattice polytope
          fiber.

        OUTPUT:

        A generator yielding the distinct lattice polytope fibers of
        given dimension.

        EXAMPLES::

            sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
            sage: p = LatticePolytope_PPL((-9,-6,-1,-1),(0,0,0,1),(0,0,1,0),(0,1,0,0),(1,0,0,0))
            sage: list( p.fibration_generator(2) )
            [A 2-dimensional lattice polytope in ZZ^4 with 3 vertices]
        """
        assert self.is_full_dimensional()
        codim = self.space_dimension() - dim
        # "points" are the potential vertices of the fiber. They are
        # in the $codim$-skeleton of the polytope, which is contained
        # in the points that saturate at least $dim$ equations.
        points = [ p for p in self._integral_points_saturating() if len(p[1])>=dim ]
        points = sorted(points, key=lambda x:len(x[1]))

        # iterate over point combinations subject to all points being on one facet.
        def point_combinations_iterator(n, i0=0, saturated=None):
            for i in range(i0, len(points)):
                p, ieqs = points[i]
                if saturated is None:
                    saturated_ieqs = ieqs
                else:
                    saturated_ieqs = saturated.intersection(ieqs)
                if len(saturated_ieqs)==0:
                    continue
                if n == 1:
                    yield [i]
                else:
                    for c in point_combinations_iterator(n-1, i+1, saturated_ieqs):
                        yield [i] + c

        point_lines = [ line(Linear_Expression(p[0].list(),0)) for p in points ]
        origin = point()
        fibers = set()
        gs = Generator_System()
        for indices in point_combinations_iterator(dim):
            gs.clear()
            gs.insert(origin)
            for i in indices:
                gs.insert(point_lines[i])
            plane = C_Polyhedron(gs)
            if plane.affine_dimension() != dim:
                continue
            plane.intersection_assign(self)
            if (not self.is_full_dimensional()) and (plane.affine_dimension() != dim):
                continue
            try:
                fiber = LatticePolytope_PPL(plane)
            except TypeError:   # not a lattice polytope
                continue
            fiber_vertices = tuple(sorted(fiber.vertices()))
            if fiber_vertices not in fibers:
                yield fiber
                fibers.update([fiber_vertices])
Пример #7
0
def LatticePolytope_PPL(*args):
    """
    Construct a new instance of the PPL-based lattice polytope class.

    EXAMPLES::

        sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
        sage: LatticePolytope_PPL((0,0),(1,0),(0,1))
        A 2-dimensional lattice polytope in ZZ^2 with 3 vertices

        sage: from sage.libs.ppl import point, Generator_System, C_Polyhedron, Linear_Expression, Variable
        sage: p = point(Linear_Expression([2,3],0));  p
        point(2/1, 3/1)
        sage: LatticePolytope_PPL(p)
        A 0-dimensional lattice polytope in ZZ^2 with 1 vertex

        sage: P = C_Polyhedron(Generator_System(p));  P
        A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point
        sage: LatticePolytope_PPL(P)
        A 0-dimensional lattice polytope in ZZ^2 with 1 vertex

    A ``TypeError`` is raised if the arguments do not specify a lattice polytope::

        sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
        sage: LatticePolytope_PPL((0,0),(1/2,1))
        Traceback (most recent call last):
        ...
        TypeError: no conversion of this rational to integer

        sage: from sage.libs.ppl import point, Generator_System, C_Polyhedron, Linear_Expression, Variable
        sage: p = point(Linear_Expression([2,3],0), 5);  p
        point(2/5, 3/5)
        sage: LatticePolytope_PPL(p)
        Traceback (most recent call last):
         ...
        TypeError: generator is not a lattice polytope generator

        sage: P = C_Polyhedron(Generator_System(p));  P
        A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point
        sage: LatticePolytope_PPL(P)
        Traceback (most recent call last):
        ...
        TypeError: polyhedron has non-integral generators
    """
    polytope_class = LatticePolytope_PPL_class
    if len(args)==1 and isinstance(args[0], C_Polyhedron):
        polyhedron = args[0]
        polytope_class = _class_for_LatticePolytope(polyhedron.space_dimension())
        if not all(p.is_point() and p.divisor().is_one() for p in polyhedron.generators()):
            raise TypeError('polyhedron has non-integral generators')
        return polytope_class(polyhedron)
    if len(args)==1 \
            and isinstance(args[0], (list, tuple)) \
            and isinstance(args[0][0], (list,tuple)):
        vertices = args[0]
    else:
        vertices = args
    gs = Generator_System()
    for v in vertices:
        if isinstance(v, Generator):
            if (not v.is_point()) or (not v.divisor().is_one()):
                raise TypeError('generator is not a lattice polytope generator')
            gs.insert(v)
        else:
            gs.insert(point(Linear_Expression(v, 0)))
    if not gs.empty():
        dim = Generator_System_iterator(gs).next().space_dimension()
        polytope_class = _class_for_LatticePolytope(dim)
    return polytope_class(gs)
Пример #8
0
    def _init_from_Vrepresentation(self, vertices, rays, lines, minimize=True, verbose=False):
        """
        Construct polyhedron from V-representation data.

        INPUT:

        - ``vertices`` -- list of point. Each point can be specified
           as any iterable container of
           :meth:`~sage.geometry.polyhedron.base.base_ring` elements.

        - ``rays`` -- list of rays. Each ray can be specified as any
          iterable container of
          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.

        - ``lines`` -- list of lines. Each line can be specified as
          any iterable container of
          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.

        - ``verbose`` -- boolean (default: ``False``). Whether to print
          verbose output for debugging purposes.

        EXAMPLES::

            sage: p = Polyhedron(backend='ppl')
            sage: from sage.geometry.polyhedron.backend_ppl import Polyhedron_ppl
            sage: Polyhedron_ppl._init_from_Vrepresentation(p, [], [], [])
        """
        gs = Generator_System()
        if vertices is None: vertices = []
        for v in vertices:
            d = LCM_list([denominator(v_i) for v_i in v])
            if d.is_one():
                gs.insert(point(Linear_Expression(v, 0)))
            else:
                dv = [ d*v_i for v_i in v ]
                gs.insert(point(Linear_Expression(dv, 0), d))
        if rays is None: rays = []
        for r in rays:
            d = LCM_list([denominator(r_i) for r_i in r])
            if d.is_one():
                gs.insert(ray(Linear_Expression(r, 0)))
            else:
                dr = [ d*r_i for r_i in r ]
                gs.insert(ray(Linear_Expression(dr, 0)))
        if lines is None: lines = []
        for l in lines:
            d = LCM_list([denominator(l_i) for l_i in l])
            if d.is_one():
                gs.insert(line(Linear_Expression(l, 0)))
            else:
                dl = [ d*l_i for l_i in l ]
                gs.insert(line(Linear_Expression(dl, 0)))
        if gs.empty():
            self._ppl_polyhedron = C_Polyhedron(self.ambient_dim(), 'empty')
        else:
            self._ppl_polyhedron = C_Polyhedron(gs)
        self._init_Vrepresentation_from_ppl(minimize)
        self._init_Hrepresentation_from_ppl(minimize)
Пример #9
0
def LatticePolytope_PPL(*args):
    """
    Construct a new instance of the PPL-based lattice polytope class.

    EXAMPLES::

        sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
        sage: LatticePolytope_PPL((0,0),(1,0),(0,1))
        A 2-dimensional lattice polytope in ZZ^2 with 3 vertices

        sage: from sage.libs.ppl import point, Generator_System, C_Polyhedron, Linear_Expression, Variable
        sage: p = point(Linear_Expression([2,3],0));  p
        point(2/1, 3/1)
        sage: LatticePolytope_PPL(p)
        A 0-dimensional lattice polytope in ZZ^2 with 1 vertex

        sage: P = C_Polyhedron(Generator_System(p));  P
        A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point
        sage: LatticePolytope_PPL(P)
        A 0-dimensional lattice polytope in ZZ^2 with 1 vertex

    A ``TypeError`` is raised if the arguments do not specify a lattice polytope::

        sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
        sage: LatticePolytope_PPL((0,0),(1/2,1))
        Traceback (most recent call last):
        ...
        TypeError: no conversion of this rational to integer

        sage: from sage.libs.ppl import point, Generator_System, C_Polyhedron, Linear_Expression, Variable
        sage: p = point(Linear_Expression([2,3],0), 5);  p
        point(2/5, 3/5)
        sage: LatticePolytope_PPL(p)
        Traceback (most recent call last):
        ...
        TypeError: The generator is not a lattice polytope generator.

        sage: P = C_Polyhedron(Generator_System(p));  P
        A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point
        sage: LatticePolytope_PPL(P)
        Traceback (most recent call last):
        ...
        TypeError: The polyhedron has non-integral generators.
    """
    if len(args)==1 and isinstance(args[0], C_Polyhedron):
        polyhedron = args[0]
        #if not polyhedron.is_bounded():
        #    raise TypeError('The polyhedron is unbounded.')
        if not all(p.is_point() and p.divisor().is_one() for p in polyhedron.generators()):
            raise TypeError('The polyhedron has non-integral generators.')
        return LatticePolytope_PPL_class(polyhedron)
    if len(args)==1:
        vertices = args[0]
        try:
            return LatticePolytope_PPL(*vertices)
        except TypeError:
            pass
    vertices = args
    gs = Generator_System()
    for v in vertices:
        if isinstance(v, Generator):
            if (not v.is_point()) or (not v.divisor().is_one()):
                raise TypeError('The generator is not a lattice polytope generator.')
            gs.insert(v)
        else:
            gs.insert(point(Linear_Expression(v, 0)))
    return LatticePolytope_PPL_class(gs)