def plot(self,
label_elements=True,
element_labels=None,
label_font_size=12,
label_font_color='black',
layout="acyclic",
**kwds):
"""
Returns a Graphics object corresponding to the Hasse diagram.
EXAMPLES::
sage: uc = [[2,3], [], [1], [1], [1], [3,4]]
sage: elm_lbls = Permutations(3).list()
sage: P = Poset(uc,elm_lbls)
sage: H = P._hasse_diagram
sage: levels = H.level_sets()
sage: heights = dict([[i, levels[i]] for i in range(len(levels))])
sage: type(H.plot(label_elements=True))
<class 'sage.plot.graphics.Graphics'>
::
sage: P = Posets.SymmetricGroupBruhatIntervalPoset([0,1,2,3], [2,3,0,1])
sage: P._hasse_diagram.plot()
"""
# Set element_labels to default to the vertex set.
if element_labels is None:
element_labels = range(self.num_verts())
# Create the underlying graph.
graph = DiGraph(self)
graph.relabel(element_labels)
return graph.plot(layout=layout, **kwds)
def plot(self, label_elements=True, element_labels=None,
label_font_size=12,label_font_color='black', layout = "acyclic", **kwds):
"""
Returns a Graphics object corresponding to the Hasse diagram.
EXAMPLES::
sage: uc = [[2,3], [], [1], [1], [1], [3,4]]
sage: elm_lbls = Permutations(3).list()
sage: P = Poset(uc,elm_lbls)
sage: H = P._hasse_diagram
sage: levels = H.level_sets()
sage: heights = dict([[i, levels[i]] for i in range(len(levels))])
sage: type(H.plot(label_elements=True))
<class 'sage.plot.graphics.Graphics'>
::
sage: P = Posets.SymmetricGroupBruhatIntervalPoset([0,1,2,3], [2,3,0,1])
sage: P._hasse_diagram.plot()
"""
# Set element_labels to default to the vertex set.
if element_labels is None:
element_labels = range(self.num_verts())
# Create the underlying graph.
graph = DiGraph(self)
graph.relabel(element_labels)
return graph.plot(layout = layout, **kwds)
def Hasse_diagram_from_incidences(atom_to_coatoms, coatom_to_atoms,
face_constructor=None,
required_atoms=None,
key = None,
**kwds):
r"""
Compute the Hasse diagram of an atomic and coatomic lattice.
INPUT:
- ``atom_to_coatoms`` -- list, ``atom_to_coatom[i]`` should list all
coatoms over the ``i``-th atom;
- ``coatom_to_atoms`` -- list, ``coatom_to_atom[i]`` should list all
atoms under the ``i``-th coatom;
- ``face_constructor`` -- function or class taking as the first two
arguments sorted :class:`tuple` of integers and any keyword arguments.
It will be called to construct a face over atoms passed as the first
argument and under coatoms passed as the second argument. Default
implementation will just return these two tuples as a tuple;
- ``required_atoms`` -- list of atoms (default:None). Each
non-empty "face" requires at least on of the specified atoms
present. Used to ensure that each face has a vertex.
- ``key`` -- any hashable value (default: None). It is passed down
to :class:`~sage.combinat.posets.posets.FinitePoset`.
- all other keyword arguments will be passed to ``face_constructor`` on
each call.
OUTPUT:
- :class:`finite poset <sage.combinat.posets.posets.FinitePoset>` with
elements constructed by ``face_constructor``.
.. NOTE::
In addition to the specified partial order, finite posets in Sage have
internal total linear order of elements which extends the partial one.
This function will try to make this internal order to start with the
bottom and atoms in the order corresponding to ``atom_to_coatoms`` and
to finish with coatoms in the order corresponding to
``coatom_to_atoms`` and the top. This may not be possible if atoms and
coatoms are the same, in which case the preference is given to the
first list.
ALGORITHM:
The detailed description of the used algorithm is given in [KP2002]_.
The code of this function follows the pseudo-code description in the
section 2.5 of the paper, although it is mostly based on frozen sets
instead of sorted lists - this makes the implementation easier and should
not cost a big performance penalty. (If one wants to make this function
faster, it should be probably written in Cython.)
While the title of the paper mentions only polytopes, the algorithm (and
the implementation provided here) is applicable to any atomic and coatomic
lattice if both incidences are given, see Section 3.4.
In particular, this function can be used for strictly convex cones and
complete fans.
REFERENCES:
.. [KP2002]
Volker Kaibel and Marc E. Pfetsch,
"Computing the Face Lattice of a Polytope from its Vertex-Facet
Incidences", Computational Geometry: Theory and Applications,
Volume 23, Issue 3 (November 2002), 281-290.
Available at http://portal.acm.org/citation.cfm?id=763203
and free of charge at http://arxiv.org/abs/math/0106043
AUTHORS:
- Andrey Novoseltsev (2010-05-13) with thanks to Marshall Hampton for the
reference.
EXAMPLES:
Let's construct the Hasse diagram of a lattice of subsets of {0, 1, 2}.
Our atoms are {0}, {1}, and {2}, while our coatoms are {0,1}, {0,2}, and
{1,2}. Then incidences are ::
sage: atom_to_coatoms = [(0,1), (0,2), (1,2)]
sage: coatom_to_atoms = [(0,1), (0,2), (1,2)]
and we can compute the Hasse diagram as ::
sage: L = sage.geometry.cone.Hasse_diagram_from_incidences(
... atom_to_coatoms, coatom_to_atoms)
sage: L
Finite poset containing 8 elements
sage: for level in L.level_sets(): print level
[((), (0, 1, 2))]
[((0,), (0, 1)), ((1,), (0, 2)), ((2,), (1, 2))]
[((0, 1), (0,)), ((0, 2), (1,)), ((1, 2), (2,))]
[((0, 1, 2), ())]
For more involved examples see the *source code* of
:meth:`sage.geometry.cone.ConvexRationalPolyhedralCone.face_lattice` and
:meth:`sage.geometry.fan.RationalPolyhedralFan._compute_cone_lattice`.
"""
from sage.graphs.all import DiGraph
from sage.combinat.posets.posets import FinitePoset
def default_face_constructor(atoms, coatoms, **kwds):
return (atoms, coatoms)
if face_constructor is None:
face_constructor = default_face_constructor
atom_to_coatoms = [frozenset(atc) for atc in atom_to_coatoms]
A = frozenset(range(len(atom_to_coatoms))) # All atoms
coatom_to_atoms = [frozenset(cta) for cta in coatom_to_atoms]
C = frozenset(range(len(coatom_to_atoms))) # All coatoms
# Comments with numbers correspond to steps in Section 2.5 of the article
L = DiGraph(1) # 3: initialize L
faces = dict()
atoms = frozenset()
coatoms = C
faces[atoms, coatoms] = 0
next_index = 1
Q = [(atoms, coatoms)] # 4: initialize Q with the empty face
while Q: # 5
q_atoms, q_coatoms = Q.pop() # 6: remove some q from Q
q = faces[q_atoms, q_coatoms]
# 7: compute H = {closure(q+atom) : atom not in atoms of q}
H = dict()
candidates = set(A.difference(q_atoms))
for atom in candidates:
coatoms = q_coatoms.intersection(atom_to_coatoms[atom])
atoms = A
for coatom in coatoms:
atoms = atoms.intersection(coatom_to_atoms[coatom])
H[atom] = (atoms, coatoms)
# 8: compute the set G of minimal sets in H
minimals = set([])
while candidates:
candidate = candidates.pop()
atoms = H[candidate][0]
if atoms.isdisjoint(candidates) and atoms.isdisjoint(minimals):
minimals.add(candidate)
# Now G == {H[atom] : atom in minimals}
for atom in minimals: # 9: for g in G:
g_atoms, g_coatoms = H[atom]
if not required_atoms is None:
if g_atoms.isdisjoint(required_atoms):
continue
if (g_atoms, g_coatoms) in faces:
g = faces[g_atoms, g_coatoms]
else: # 11: if g was newly created
g = next_index
faces[g_atoms, g_coatoms] = g
next_index += 1
Q.append((g_atoms, g_coatoms)) # 12
L.add_edge(q, g) # 14
# End of algorithm, now construct a FinitePoset.
# In principle, it is recommended to use Poset or in this case perhaps
# even LatticePoset, but it seems to take several times more time
# than the above computation, makes unnecessary copies, and crashes.
# So for now we will mimic the relevant code from Poset.
# Enumeration of graph vertices must be a linear extension of the poset
new_order = L.topological_sort()
# Make sure that coatoms are in the end in proper order
tail = [faces[atoms, frozenset([coatom])]
for coatom, atoms in enumerate(coatom_to_atoms)]
tail.append(faces[A, frozenset()])
new_order = [n for n in new_order if n not in tail] + tail
# Make sure that atoms are in the beginning in proper order
head = [0] # We know that the empty face has index 0
head.extend(faces[frozenset([atom]), coatoms]
for atom, coatoms in enumerate(atom_to_coatoms)
if required_atoms is None or atom in required_atoms)
new_order = head + [n for n in new_order if n not in head]
# "Invert" this list to a dictionary
labels = dict()
for new, old in enumerate(new_order):
labels[old] = new
L.relabel(labels)
# Construct the actual poset elements
elements = [None] * next_index
for face, index in faces.items():
atoms, coatoms = face
elements[labels[index]] = face_constructor(
tuple(sorted(atoms)), tuple(sorted(coatoms)), **kwds)
return FinitePoset(L, elements, key = key)
