def lift_cross_ratios(A, lift_map=None):
r"""
Return a matrix which arises from the given matrix by lifting cross ratios.
INPUT:
- ``A`` -- a matrix over a ring ``source_ring``.
- ``lift_map`` -- a python dictionary, mapping each cross ratio of ``A`` to some element
of a target ring, and such that ``lift_map[source_ring(1)] = target_ring(1)``.
OUTPUT:
- ``Z`` -- a matrix over the ring ``target_ring``.
The intended use of this method is to create a (reduced) matrix representation of a
matroid ``M`` over a ring ``target_ring``, given a (reduced) matrix representation of
``A`` of ``M`` over a ring ``source_ring`` and a map ``lift_map`` from ``source_ring``
to ``target_ring``.
This method will create a unique candidate representation ``Z``, but will not verify
if ``Z`` is indeed a representation of ``M``. However, this is guaranteed if the
conditions of the lift theorem (see [PvZ2010]_) hold for the lift map in combination with
the matrix ``A``.
For a lift map `f` and a matrix `A` these conditions are as follows. First of all
`f: S \rightarrow T`, where `S` is a set of invertible elements of the source ring and
`T` is a set of invertible elements of the target ring. The matrix `A` has entries
from the source ring, and each cross ratio of `A` is contained in `S`. Moreover:
- `1 \in S`, `1 \in T`;
- for all `x \in S`: `f(x) = 1` if and only if `x = 1`;
- for all `x, y \in S`: if `x + y = 0` then `f(x) + f(y) = 0`;
- for all `x, y \in S`: if `x + y = 1` then `f(x) + f(y) = 1`;
- for all `x, y, z \in S`: if `xy = z` then `f(x)f(y) = f(z)`.
Any ring homomorphism `h: P \rightarrow R` induces a lift map from the set of units `S` of
`P` to the set of units `T` of `R`. There exist lift maps which do not arise in
this manner. Several such maps can be created by the function
:meth:`lift_map() <sage.matroids.utilities.lift_map>`.
.. SEEALSO::
:meth:`lift_map() <sage.matroids.utilities.lift_map>`
EXAMPLES::
sage: from sage.matroids.advanced import lift_cross_ratios, lift_map, LinearMatroid
sage: R = GF(7)
sage: to_sixth_root_of_unity = lift_map('sru')
sage: A = Matrix(R, [[1, 0, 6, 1, 2],[6, 1, 0, 0, 1],[0, 6, 3, 6, 0]])
sage: A
[1 0 6 1 2]
[6 1 0 0 1]
[0 6 3 6 0]
sage: Z = lift_cross_ratios(A, to_sixth_root_of_unity)
sage: Z
[ 1 0 1 1 1]
[ 1 1 0 0 z]
[ 0 z - 1 1 -z + 1 0]
sage: M = LinearMatroid(reduced_matrix = A)
sage: sorted(M.cross_ratios())
[3, 5]
sage: N = LinearMatroid(reduced_matrix = Z)
sage: sorted(N.cross_ratios())
[-z + 1, z]
sage: M.is_isomorphism(N, {e:e for e in M.groundset()})
True
"""
for s, t in iteritems(lift_map):
source_ring = s.parent()
target_ring = t.parent()
break
plus_one1 = source_ring(1)
minus_one1 = source_ring(-1)
plus_one2 = target_ring(1)
minus_one2 = target_ring(-1)
G = Graph([((r, 0), (c, 1), (r, c)) for r, c in A.nonzero_positions()])
# write the entries of (a scaled version of) A as products of cross ratios of A
T = set()
for C in G.connected_components():
T.update(G.subgraph(C).min_spanning_tree())
# - fix a tree of the support graph G to units (= empty dict, product of 0 terms)
F = {entry[2]: dict() for entry in T}
W = set(G.edges()) - set(T)
H = G.subgraph(edges=T)
while W:
# - find an edge in W to process, closing a circuit in H which is induced in G
edge = W.pop()
path = H.shortest_path(edge[0], edge[1])
retry = True
while retry:
retry = False
for edge2 in W:
if edge2[0] in path and edge2[1] in path:
W.add(edge)
edge = edge2
W.remove(edge)
path = H.shortest_path(edge[0], edge[1])
retry = True
break
entry = edge[2]
entries = []
for i in range(len(path) - 1):
v = path[i]
w = path[i + 1]
if v[1] == 0:
entries.append((v[0], w[0]))
else:
entries.append((w[0], v[0]))
# - compute the cross ratio `cr` of this whirl
cr = source_ring(A[entry])
div = True
for entry2 in entries:
if div:
cr = cr / A[entry2]
else:
cr = cr * A[entry2]
div = not div
monomial = dict()
if len(path) % 4 == 0:
if not cr == plus_one1:
monomial[cr] = 1
else:
cr = -cr
if not cr == plus_one1:
monomial[cr] = 1
if minus_one1 in monomial:
monomial[minus_one1] = monomial[minus_one1] + 1
else:
monomial[minus_one1] = 1
if cr != plus_one1 and not cr in lift_map:
raise ValueError("Input matrix has a cross ratio " + str(cr) +
", which is not in the lift_map")
# - write the entry as a product of cross ratios of A
div = True
for entry2 in entries:
if div:
for cr, degree in iteritems(F[entry2]):
if cr in monomial:
monomial[cr] = monomial[cr] + degree
else:
monomial[cr] = degree
else:
for cr, degree in iteritems(F[entry2]):
if cr in monomial:
monomial[cr] = monomial[cr] - degree
else:
monomial[cr] = -degree
div = not div
F[entry] = monomial
# - current edge is done, can be used in next iteration
H.add_edge(edge)
# compute each entry of Z as the product of lifted cross ratios
Z = Matrix(target_ring, A.nrows(), A.ncols())
for entry, monomial in iteritems(F):
Z[entry] = plus_one2
for cr, degree in iteritems(monomial):
if cr == minus_one1:
Z[entry] = Z[entry] * (minus_one2**degree)
else:
Z[entry] = Z[entry] * (lift_map[cr]**degree)
return Z
def lift_cross_ratios(A, lift_map = None):
r"""
Return a matrix which arises from the given matrix by lifting cross ratios.
INPUT:
- ``A`` -- a matrix over a ring ``source_ring``.
- ``lift_map`` -- a python dictionary, mapping each cross ratio of ``A`` to some element
of a target ring, and such that ``lift_map[source_ring(1)] = target_ring(1)``.
OUTPUT:
- ``Z`` -- a matrix over the ring ``target_ring``.
The intended use of this method is to create a (reduced) matrix representation of a
matroid ``M`` over a ring ``target_ring``, given a (reduced) matrix representation of
``A`` of ``M`` over a ring ``source_ring`` and a map ``lift_map`` from ``source_ring``
to ``target_ring``.
This method will create a unique candidate representation ``Z``, but will not verify
if ``Z`` is indeed a representation of ``M``. However, this is guaranteed if the
conditions of the lift theorem (see [PvZ2010]_) hold for the lift map in combination with
the matrix ``A``.
For a lift map `f` and a matrix `A` these conditions are as follows. First of all
`f: S \rightarrow T`, where `S` is a set of invertible elements of the source ring and
`T` is a set of invertible elements of the target ring. The matrix `A` has entries
from the source ring, and each cross ratio of `A` is contained in `S`. Moreover:
- `1 \in S`, `1 \in T`;
- for all `x \in S`: `f(x) = 1` if and only if `x = 1`;
- for all `x, y \in S`: if `x + y = 0` then `f(x) + f(y) = 0`;
- for all `x, y \in S`: if `x + y = 1` then `f(x) + f(y) = 1`;
- for all `x, y, z \in S`: if `xy = z` then `f(x)f(y) = f(z)`.
Any ring homomorphism `h: P \rightarrow R` induces a lift map from the set of units `S` of
`P` to the set of units `T` of `R`. There exist lift maps which do not arise in
this manner. Several such maps can be created by the function
:meth:`lift_map() <sage.matroids.utilities.lift_map>`.
.. SEEALSO::
:meth:`lift_map() <sage.matroids.utilities.lift_map>`
EXAMPLES::
sage: from sage.matroids.advanced import lift_cross_ratios, lift_map, LinearMatroid
sage: R = GF(7)
sage: to_sixth_root_of_unity = lift_map('sru')
sage: A = Matrix(R, [[1, 0, 6, 1, 2],[6, 1, 0, 0, 1],[0, 6, 3, 6, 0]])
sage: A
[1 0 6 1 2]
[6 1 0 0 1]
[0 6 3 6 0]
sage: Z = lift_cross_ratios(A, to_sixth_root_of_unity)
sage: Z
[ 1 0 1 1 1]
[ 1 1 0 0 z]
[ 0 z - 1 1 -z + 1 0]
sage: M = LinearMatroid(reduced_matrix = A)
sage: sorted(M.cross_ratios())
[3, 5]
sage: N = LinearMatroid(reduced_matrix = Z)
sage: sorted(N.cross_ratios())
[-z + 1, z]
sage: M.is_isomorphism(N, {e:e for e in M.groundset()})
True
"""
for s, t in iteritems(lift_map):
source_ring = s.parent()
target_ring = t.parent()
break
plus_one1 = source_ring(1)
minus_one1 = source_ring(-1)
plus_one2 = target_ring(1)
minus_one2 = target_ring(-1)
G = Graph([((r,0),(c,1),(r,c)) for r,c in A.nonzero_positions()])
# write the entries of (a scaled version of) A as products of cross ratios of A
T = set()
for C in G.connected_components():
T.update(G.subgraph(C).min_spanning_tree())
# - fix a tree of the support graph G to units (= empty dict, product of 0 terms)
F = {entry[2]: dict() for entry in T}
W = set(G.edges()) - set(T)
H = G.subgraph(edges = T)
while W:
# - find an edge in W to process, closing a circuit in H which is induced in G
edge = W.pop()
path = H.shortest_path(edge[0], edge[1])
retry = True
while retry:
retry = False
for edge2 in W:
if edge2[0] in path and edge2[1] in path:
W.add(edge)
edge = edge2
W.remove(edge)
path = H.shortest_path(edge[0], edge[1])
retry = True
break
entry = edge[2]
entries = []
for i in range(len(path) - 1):
v = path[i]
w = path[i+1]
if v[1] == 0:
entries.append((v[0],w[0]))
else:
entries.append((w[0],v[0]))
# - compute the cross ratio `cr` of this whirl
cr = source_ring(A[entry])
div = True
for entry2 in entries:
if div:
cr = cr/A[entry2]
else:
cr = cr* A[entry2]
div = not div
monomial = dict()
if len(path) % 4 == 0:
if not cr == plus_one1:
monomial[cr] = 1
else:
cr = -cr
if not cr ==plus_one1:
monomial[cr] = 1
if minus_one1 in monomial:
monomial[minus_one1] = monomial[minus_one1] + 1
else:
monomial[minus_one1] = 1
if cr != plus_one1 and not cr in lift_map:
raise ValueError("Input matrix has a cross ratio "+str(cr)+", which is not in the lift_map")
# - write the entry as a product of cross ratios of A
div = True
for entry2 in entries:
if div:
for cr, degree in iteritems(F[entry2]):
if cr in monomial:
monomial[cr] = monomial[cr]+ degree
else:
monomial[cr] = degree
else:
for cr, degree in iteritems(F[entry2]):
if cr in monomial:
monomial[cr] = monomial[cr] - degree
else:
monomial[cr] = -degree
div = not div
F[entry] = monomial
# - current edge is done, can be used in next iteration
H.add_edge(edge)
# compute each entry of Z as the product of lifted cross ratios
Z = Matrix(target_ring, A.nrows(), A.ncols())
for entry, monomial in iteritems(F):
Z[entry] = plus_one2
for cr,degree in iteritems(monomial):
if cr == minus_one1:
Z[entry] = Z[entry] * (minus_one2**degree)
else:
Z[entry] = Z[entry] * (lift_map[cr]**degree)
return Z
def Matroid(*args, **kwds):
r"""
Construct a matroid.
Matroids are combinatorial structures that capture the abstract properties
of (linear/algebraic/...) dependence. Formally, a matroid is a pair
`M = (E, I)` of a finite set `E`, the *groundset*, and a collection of
subsets `I`, the independent sets, subject to the following axioms:
* `I` contains the empty set
* If `X` is a set in `I`, then each subset of `X` is in `I`
* If two subsets `X`, `Y` are in `I`, and `|X| > |Y|`, then there exists
`x \in X - Y` such that `Y + \{x\}` is in `I`.
See the :wikipedia:`Wikipedia article on matroids <Matroid>` for more
theory and examples. Matroids can be obtained from many types of
mathematical structures, and Sage supports a number of them.
There are two main entry points to Sage's matroid functionality. For
built-in matroids, do the following:
* Within a Sage session, type "matroids." (Do not press "Enter", and do
not forget the final period ".")
* Hit "tab".
You will see a list of methods which will construct matroids. For
example::
sage: F7 = matroids.named_matroids.Fano()
sage: len(F7.nonspanning_circuits())
7
or::
sage: U36 = matroids.Uniform(3, 6)
sage: U36.equals(U36.dual())
True
To define your own matroid, use the function ``Matroid()``.
This function attempts to interpret its arguments to create an appropriate
matroid. The following named arguments are supported:
INPUT:
- ``groundset`` -- If provided, the groundset of the matroid.
If not provided, the function attempts to determine a groundset from the
data.
- ``bases`` -- The list of bases (maximal independent sets) of the
matroid.
- ``independent_sets`` -- The list of independent sets of the matroid.
- ``circuits`` -- The list of circuits of the matroid.
- ``graph`` -- A graph, whose edges form the elements of the matroid.
- ``matrix`` -- A matrix representation of the matroid.
- ``reduced_matrix`` -- A reduced representation of the matroid: if
``reduced_matrix = A``
then the matroid is represented by `[I\ \ A]` where `I` is an
appropriately sized identity matrix.
- ``rank_function`` -- A function that computes the rank of each subset.
Can only be provided together with a groundset.
- ``circuit_closures`` -- Either a list of tuples ``(k, C)`` with ``C``
the closure of a circuit, and ``k`` the rank of ``C``, or a dictionary
``D`` with ``D[k]`` the set of closures of rank-``k`` circuits.
- ``matroid`` -- An object that is already a matroid. Useful only with the
``regular`` option.
Up to two unnamed arguments are allowed.
- One unnamed argument, no named arguments other than ``regular`` -- the
input should be either a graph, or a matrix, or a list of independent
sets containing all bases, or a matroid.
- Two unnamed arguments: the first is the groundset, the second a graph,
or a matrix, or a list of independent sets containing all bases, or a
matroid.
- One unnamed argument, at least one named argument: the unnamed argument
is the groundset, the named argument is as above (but must be different
from ``groundset``).
The examples section details how each of the input types deals with
explicit or implicit groundset arguments.
OPTIONS:
- ``regular`` -- (default: ``False``) boolean. If ``True``,
output a
:class:`RegularMatroid <sage.matroids.linear_matroid.RegularMatroid>`
instance such that, *if* the input defines a valid regular matroid, then
the output represents this matroid. Note that this option can be
combined with any type of input.
- ``ring`` -- any ring. If provided, and the input is a ``matrix`` or
``reduced_matrix``, output will be a linear matroid over the ring or
field ``ring``.
- ``field`` -- any field. Same as ``ring``, but only fields are allowed.
- ``check`` -- (default: ``True``) boolean. If ``True`` and
``regular`` is true, the output is checked to make sure it is a valid
regular matroid.
.. WARNING::
Except for regular matroids, the input is not checked for validity. If
your data does not correspond to an actual matroid, the behavior of
the methods is undefined and may cause strange errors. To ensure you
have a matroid, run
:meth:`M.is_valid() <sage.matroids.matroid.Matroid.is_valid>`.
.. NOTE::
The ``Matroid()`` method will return instances of type
:class:`BasisMatroid <sage.matroids.basis_matroid.BasisMatroid>`,
:class:`CircuitClosuresMatroid <sage.matroids.circuit_closures_matroid.CircuitClosuresMatroid>`,
:class:`LinearMatroid <sage.matroids.linear_matroid.LinearMatroid>`,
:class:`BinaryMatroid <sage.matroids.linear_matroid.LinearMatroid>`,
:class:`TernaryMatroid <sage.matroids.linear_matroid.LinearMatroid>`,
:class:`QuaternaryMatroid <sage.matroids.linear_matroid.LinearMatroid>`,
:class:`RegularMatroid <sage.matroids.linear_matroid.LinearMatroid>`, or
:class:`RankMatroid <sage.matroids.rank_matroid.RankMatroid>`. To
import these classes (and other useful functions) directly into Sage's
main namespace, type::
sage: from sage.matroids.advanced import *
See :mod:`sage.matroids.advanced <sage.matroids.advanced>`.
EXAMPLES:
Note that in these examples we will often use the fact that strings are
iterable in these examples. So we type ``'abcd'`` to denote the list
``['a', 'b', 'c', 'd']``.
#. List of bases:
All of the following inputs are allowed, and equivalent::
sage: M1 = Matroid(groundset='abcd', bases=['ab', 'ac', 'ad',
....: 'bc', 'bd', 'cd'])
sage: M2 = Matroid(bases=['ab', 'ac', 'ad', 'bc', 'bd', 'cd'])
sage: M3 = Matroid(['ab', 'ac', 'ad', 'bc', 'bd', 'cd'])
sage: M4 = Matroid('abcd', ['ab', 'ac', 'ad', 'bc', 'bd', 'cd'])
sage: M5 = Matroid('abcd', bases=[['a', 'b'], ['a', 'c'],
....: ['a', 'd'], ['b', 'c'],
....: ['b', 'd'], ['c', 'd']])
sage: M1 == M2
True
sage: M1 == M3
True
sage: M1 == M4
True
sage: M1 == M5
True
We do not check if the provided input forms an actual matroid::
sage: M1 = Matroid(groundset='abcd', bases=['ab', 'cd'])
sage: M1.full_rank()
2
sage: M1.is_valid()
False
Bases may be repeated::
sage: M1 = Matroid(['ab', 'ac'])
sage: M2 = Matroid(['ab', 'ac', 'ab'])
sage: M1 == M2
True
#. List of independent sets:
::
sage: M1 = Matroid(groundset='abcd',
....: independent_sets=['', 'a', 'b', 'c', 'd', 'ab',
....: 'ac', 'ad', 'bc', 'bd', 'cd'])
We only require that the list of independent sets contains each basis
of the matroid; omissions of smaller independent sets and
repetitions are allowed::
sage: M1 = Matroid(bases=['ab', 'ac'])
sage: M2 = Matroid(independent_sets=['a', 'ab', 'b', 'ab', 'a',
....: 'b', 'ac'])
sage: M1 == M2
True
#. List of circuits:
::
sage: M1 = Matroid(groundset='abc', circuits=['bc'])
sage: M2 = Matroid(bases=['ab', 'ac'])
sage: M1 == M2
True
A matroid specified by a list of circuits gets converted to a
:class:`BasisMatroid <sage.matroids.basis_matroid.BasisMatroid>`
internally::
sage: M = Matroid(groundset='abcd', circuits=['abc', 'abd', 'acd',
....: 'bcd'])
sage: type(M)
<type 'sage.matroids.basis_matroid.BasisMatroid'>
Strange things can happen if the input does not satisfy the circuit
axioms, and these are not always caught by the
:meth:`is_valid() <sage.matroids.matroid.Matroid.is_valid>` method. So
always check whether your input makes sense!
::
sage: M = Matroid('abcd', circuits=['ab', 'acd'])
sage: M.is_valid()
True
sage: [sorted(C) for C in M.circuits()]
[['a']]
#. Graph:
Sage has great support for graphs, see :mod:`sage.graphs.graph`.
::
sage: G = graphs.PetersenGraph()
sage: Matroid(G)
Regular matroid of rank 9 on 15 elements with 2000 bases
Note: if a groundset is specified, we assume it is in the same order
as
:meth:`G.edge_iterator() <sage.graphs.generic_graph.GenericGraph.edge_iterator>`
provides::
sage: G = Graph([(0, 1), (0, 2), (0, 2), (1, 2)])
sage: M = Matroid('abcd', G)
sage: M.rank(['b', 'c'])
1
If no groundset is provided, we attempt to use the edge labels::
sage: G = Graph([(0, 1, 'a'), (0, 2, 'b'), (1, 2, 'c')])
sage: M = Matroid(G)
sage: sorted(M.groundset())
['a', 'b', 'c']
If no edge labels are present and the graph is simple, we use the
tuples ``(i, j)`` of endpoints. If that fails, we simply use a list
``[0..m-1]`` ::
sage: G = Graph([(0, 1), (0, 2), (1, 2)])
sage: M = Matroid(G)
sage: sorted(M.groundset())
[(0, 1), (0, 2), (1, 2)]
sage: G = Graph([(0, 1), (0, 2), (0, 2), (1, 2)])
sage: M = Matroid(G)
sage: sorted(M.groundset())
[0, 1, 2, 3]
When the ``graph`` keyword is used, a variety of inputs can be
converted to a graph automatically. The following uses a graph6 string
(see the :class:`Graph <sage.graphs.graph.Graph>` method's
documentation)::
sage: Matroid(graph=':I`AKGsaOs`cI]Gb~')
Regular matroid of rank 9 on 17 elements with 4004 bases
However, this method is no more clever than ``Graph()``::
sage: Matroid(graph=41/2)
Traceback (most recent call last):
...
ValueError: input does not seem to represent a graph.
#. Matrix:
The basic input is a
:mod:`Sage matrix <sage.matrix.constructor>`::
sage: A = Matrix(GF(2), [[1, 0, 0, 1, 1, 0],
....: [0, 1, 0, 1, 0, 1],
....: [0, 0, 1, 0, 1, 1]])
sage: M = Matroid(matrix=A)
sage: M.is_isomorphic(matroids.CompleteGraphic(4))
True
Various shortcuts are possible::
sage: M1 = Matroid(matrix=[[1, 0, 0, 1, 1, 0],
....: [0, 1, 0, 1, 0, 1],
....: [0, 0, 1, 0, 1, 1]], ring=GF(2))
sage: M2 = Matroid(reduced_matrix=[[1, 1, 0],
....: [1, 0, 1],
....: [0, 1, 1]], ring=GF(2))
sage: M3 = Matroid(groundset=[0, 1, 2, 3, 4, 5],
....: matrix=[[1, 1, 0], [1, 0, 1], [0, 1, 1]],
....: ring=GF(2))
sage: A = Matrix(GF(2), [[1, 1, 0], [1, 0, 1], [0, 1, 1]])
sage: M4 = Matroid([0, 1, 2, 3, 4, 5], A)
sage: M1 == M2
True
sage: M1 == M3
True
sage: M1 == M4
True
However, with unnamed arguments the input has to be a ``Matrix``
instance, or the function will try to interpret it as a set of bases::
sage: Matroid([0, 1, 2], [[1, 0, 1], [0, 1, 1]])
Traceback (most recent call last):
...
ValueError: basis has wrong cardinality.
If the groundset size equals number of rows plus number of columns, an
identity matrix is prepended. Otherwise the groundset size must equal
the number of columns::
sage: A = Matrix(GF(2), [[1, 1, 0], [1, 0, 1], [0, 1, 1]])
sage: M = Matroid([0, 1, 2], A)
sage: N = Matroid([0, 1, 2, 3, 4, 5], A)
sage: M.rank()
2
sage: N.rank()
3
We automatically create an optimized subclass, if available::
sage: Matroid([0, 1, 2, 3, 4, 5],
....: matrix=[[1, 1, 0], [1, 0, 1], [0, 1, 1]],
....: field=GF(2))
Binary matroid of rank 3 on 6 elements, type (2, 7)
sage: Matroid([0, 1, 2, 3, 4, 5],
....: matrix=[[1, 1, 0], [1, 0, 1], [0, 1, 1]],
....: field=GF(3))
Ternary matroid of rank 3 on 6 elements, type 0-
sage: Matroid([0, 1, 2, 3, 4, 5],
....: matrix=[[1, 1, 0], [1, 0, 1], [0, 1, 1]],
....: field=GF(4, 'x'))
Quaternary matroid of rank 3 on 6 elements
sage: Matroid([0, 1, 2, 3, 4, 5],
....: matrix=[[1, 1, 0], [1, 0, 1], [0, 1, 1]],
....: field=GF(2), regular=True)
Regular matroid of rank 3 on 6 elements with 16 bases
Otherwise the generic LinearMatroid class is used::
sage: Matroid([0, 1, 2, 3, 4, 5],
....: matrix=[[1, 1, 0], [1, 0, 1], [0, 1, 1]],
....: field=GF(83))
Linear matroid of rank 3 on 6 elements represented over the Finite
Field of size 83
An integer matrix is automatically converted to a matrix over `\QQ`.
If you really want integers, you can specify the ring explicitly::
sage: A = Matrix([[1, 1, 0], [1, 0, 1], [0, 1, -1]])
sage: A.base_ring()
Integer Ring
sage: M = Matroid([0, 1, 2, 3, 4, 5], A)
sage: M.base_ring()
Rational Field
sage: M = Matroid([0, 1, 2, 3, 4, 5], A, ring=ZZ)
sage: M.base_ring()
Integer Ring
#. Rank function:
Any function mapping subsets to integers can be used as input::
sage: def f(X):
....: return min(len(X), 2)
....:
sage: M = Matroid('abcd', rank_function=f)
sage: M
Matroid of rank 2 on 4 elements
sage: M.is_isomorphic(matroids.Uniform(2, 4))
True
#. Circuit closures:
This is often a really concise way to specify a matroid. The usual way
is a dictionary of lists::
sage: M = Matroid(circuit_closures={3: ['edfg', 'acdg', 'bcfg',
....: 'cefh', 'afgh', 'abce', 'abdf', 'begh', 'bcdh', 'adeh'],
....: 4: ['abcdefgh']})
sage: M.equals(matroids.named_matroids.P8())
True
You can also input tuples `(k, X)` where `X` is the closure of a
circuit, and `k` the rank of `X`::
sage: M = Matroid(circuit_closures=[(2, 'abd'), (3, 'abcdef'),
....: (2, 'bce')])
sage: M.equals(matroids.named_matroids.Q6())
True
#. Matroid:
Most of the time, the matroid itself is returned::
sage: M = matroids.named_matroids.Fano()
sage: N = Matroid(M)
sage: N is M
True
But it can be useful with the ``regular`` option::
sage: M = Matroid(circuit_closures={2:['adb', 'bec', 'cfa',
....: 'def'], 3:['abcdef']})
sage: N = Matroid(M, regular=True)
sage: N
Regular matroid of rank 3 on 6 elements with 16 bases
sage: Matrix(N)
[1 0 0 1 1 0]
[0 1 0 1 1 1]
[0 0 1 0 1 1]
The ``regular`` option:
::
sage: M = Matroid(reduced_matrix=[[1, 1, 0],
....: [1, 0, 1],
....: [0, 1, 1]], regular=True)
sage: M
Regular matroid of rank 3 on 6 elements with 16 bases
sage: M.is_isomorphic(matroids.CompleteGraphic(4))
True
By default we check if the resulting matroid is actually regular. To
increase speed, this check can be skipped::
sage: M = matroids.named_matroids.Fano()
sage: N = Matroid(M, regular=True)
Traceback (most recent call last):
...
ValueError: input does not correspond to a valid regular matroid.
sage: N = Matroid(M, regular=True, check=False)
sage: N
Regular matroid of rank 3 on 7 elements with 32 bases
sage: N.is_valid()
False
Sometimes the output is regular, but represents a different matroid
from the one you intended::
sage: M = Matroid(Matrix(GF(3), [[1, 0, 1, 1], [0, 1, 1, 2]]))
sage: N = Matroid(Matrix(GF(3), [[1, 0, 1, 1], [0, 1, 1, 2]]),
....: regular=True)
sage: N.is_valid()
True
sage: N.is_isomorphic(M)
False
"""
# These are the valid arguments:
inputS = set([
'groundset', 'bases', 'independent_sets', 'circuits', 'graph',
'matrix', 'reduced_matrix', 'rank_function', 'circuit_closures',
'matroid'
])
# process options
if 'regular' in kwds:
want_regular = kwds['regular']
kwds.pop('regular')
else:
want_regular = False
if 'check' in kwds:
want_check = kwds['check']
kwds.pop('check')
else:
want_check = True
base_ring = None
have_field = False
if 'field' in kwds:
base_ring = kwds['field']
kwds.pop('field')
have_field = True
if 'ring' in kwds:
raise ValueError("only one of ring and field can be specified.")
try:
if not base_ring.is_field():
raise TypeError("specified ``field`` is not a field.")
except AttributeError:
raise TypeError("specified ``field`` is not a field.")
if 'ring' in kwds:
base_ring = kwds['ring']
kwds.pop('ring')
try:
if not base_ring.is_ring():
raise TypeError("specified ``ring`` is not a ring.")
except AttributeError:
raise TypeError("specified ``ring`` is not a ring.")
# Process unnamed arguments
if len(args) > 0:
if 'groundset' in kwds:
raise ValueError(
'when using unnamed arguments, groundset must be the first unnamed argument or be implicit.'
)
if len(args) > 2:
raise ValueError('at most two unnamed arguments are allowed.')
if len(args) == 2 or len(kwds) > 0:
# First argument should be the groundset
kwds['groundset'] = args[0]
# Check for unnamed data
dataindex = -1
if len(args) == 2:
dataindex = 1
if len(args) == 1 and len(kwds) == 0:
dataindex = 0
if dataindex > -1:
# One unnamed argument, no named arguments
if isinstance(args[dataindex], sage.graphs.graph.Graph):
kwds['graph'] = args[dataindex]
elif isinstance(args[dataindex], sage.matrix.matrix.Matrix):
kwds['matrix'] = args[dataindex]
elif isinstance(args[dataindex], sage.matroids.matroid.Matroid):
kwds['matroid'] = args[dataindex]
else:
kwds['independent_sets'] = args[dataindex]
# Check for multiple types of input
if len(set(kwds).difference(inputS)) > 0:
raise ValueError("unknown input argument")
if ('grondset' in kwds and len(kwds) != 2) or ('groundset' not in kwds
and len(kwds) > 1):
raise ValueError("only one type of input may be specified.")
# Bases:
if 'bases' in kwds:
if 'groundset' not in kwds:
gs = set()
for B in kwds['bases']:
gs.update(B)
kwds['groundset'] = gs
M = BasisMatroid(groundset=kwds['groundset'], bases=kwds['bases'])
# Independent sets:
if 'independent_sets' in kwds:
# Convert to list of bases first
rk = -1
bases = []
for I in kwds['independent_sets']:
if len(I) == rk:
bases.append(I)
elif len(I) > rk:
bases = [I]
rk = len(I)
if 'groundset' not in kwds:
gs = set()
for B in bases:
gs.update(B)
kwds['groundset'] = gs
M = BasisMatroid(groundset=kwds['groundset'], bases=bases)
# Circuits:
if 'circuits' in kwds:
# Convert to list of bases first
# Determine groundset (note that this cannot detect coloops)
if 'groundset' not in kwds:
gs = set()
for C in kwds['circuits']:
gs.update(C)
kwds['groundset'] = gs
# determine the rank by computing a basis
B = set(kwds['groundset'])
for C in kwds['circuits']:
I = B.intersection(C)
if len(I) >= len(C):
B.discard(I.pop())
rk = len(B)
# Construct the basis matroid of appropriate rank. Note: slow!
BB = [
frozenset(B) for B in combinations(kwds['groundset'], rk)
if not any([frozenset(C).issubset(B) for C in kwds['circuits']])
]
M = BasisMatroid(groundset=kwds['groundset'], bases=BB)
# Graphs:
if 'graph' in kwds:
# Construct the incidence matrix
# NOTE: we are not using Sage's built-in method because
# 1) we would need to fix the loops anyway
# 2) Sage will sort the columns, making it impossible to keep labels!
G = kwds['graph']
if not isinstance(G, sage.graphs.generic_graph.GenericGraph):
try:
G = Graph(G)
except (ValueError, TypeError, NetworkXError):
raise ValueError("input does not seem to represent a graph.")
V = G.vertices()
E = G.edges()
n = G.num_verts()
m = G.num_edges()
A = Matrix(ZZ, n, m, 0)
mm = 0
for i, j, k in G.edge_iterator():
A[V.index(i), mm] = -1
A[V.index(j), mm] += 1 # So loops get 0
mm += 1
# Decide on the groundset
if 'groundset' not in kwds:
# 1. Attempt to use edge labels.
sl = G.edge_labels()
if len(sl) == len(set(sl)):
kwds['groundset'] = sl
# 2. If simple, use vertex tuples
elif not G.has_multiple_edges():
kwds['groundset'] = [(i, j) for i, j, k in G.edge_iterator()]
else:
# 3. Use numbers
kwds['groundset'] = range(m)
M = RegularMatroid(matrix=A, groundset=kwds['groundset'])
want_regular = False # Save some time, since result is already regular
# Matrices:
if 'matrix' in kwds or 'reduced_matrix' in kwds:
if 'matrix' in kwds:
A = kwds['matrix']
if 'reduced_matrix' in kwds:
A = kwds['reduced_matrix']
# Fix the representation
if not isinstance(A, sage.matrix.matrix.Matrix):
try:
if base_ring is not None:
A = Matrix(base_ring, A)
else:
A = Matrix(A)
except ValueError:
raise ValueError("input does not seem to contain a matrix.")
# Fix the ring
if base_ring is not None:
if A.base_ring() != base_ring:
A = A.change_ring(base_ring)
elif A.base_ring(
) == ZZ and not want_regular: # Usually a rational matrix is intended, we presume.
A = A.change_ring(QQ)
base_ring = QQ
else:
base_ring = A.base_ring()
# Determine groundset:
if 'matrix' in kwds:
if 'groundset' in kwds:
if len(kwds['groundset']) == A.nrows() + A.ncols():
kwds['reduced_matrix'] = A
kwds.pop('matrix')
else:
if len(kwds['groundset']) != A.ncols():
raise ValueError(
"groundset size does not correspond to matrix size."
)
else:
kwds['groundset'] = range(A.ncols())
if 'reduced_matrix' in kwds:
if 'groundset' in kwds:
if len(kwds['groundset']) != A.nrows() + A.ncols():
raise ValueError(
"groundset size does not correspond to matrix size.")
else:
kwds['groundset'] = range(A.nrows() + A.ncols())
if 'matrix' in kwds:
if base_ring == GF(2):
M = BinaryMatroid(groundset=kwds['groundset'], matrix=A)
elif base_ring == GF(3):
M = TernaryMatroid(groundset=kwds['groundset'], matrix=A)
elif base_ring.is_field() and base_ring.order(
) == 4: # GF(4) can have different generators.
M = QuaternaryMatroid(groundset=kwds['groundset'], matrix=A)
else:
M = LinearMatroid(groundset=kwds['groundset'],
matrix=A,
ring=base_ring)
if 'reduced_matrix' in kwds:
if A.base_ring() == GF(2):
M = BinaryMatroid(groundset=kwds['groundset'],
reduced_matrix=A)
elif A.base_ring() == GF(3):
M = TernaryMatroid(groundset=kwds['groundset'],
reduced_matrix=A)
elif A.base_ring().is_field() and A.base_ring().order(
) == 4: # GF(4) can have different generators.
M = QuaternaryMatroid(groundset=kwds['groundset'],
reduced_matrix=A)
else:
M = LinearMatroid(groundset=kwds['groundset'],
reduced_matrix=A,
ring=base_ring)
# Rank functions:
if 'rank_function' in kwds:
if 'groundset' not in kwds:
raise ValueError(
'for rank functions, groundset needs to be specified.')
M = RankMatroid(groundset=kwds['groundset'],
rank_function=kwds['rank_function'])
# Circuit closures:
if 'circuit_closures' in kwds:
if 'groundset' not in kwds:
E = set()
if isinstance(kwds['circuit_closures'], dict):
for X in kwds['circuit_closures'].itervalues():
for Y in X:
E.update(Y)
else:
for X in kwds['circuit_closures']:
E.update(X[1])
else:
E = kwds['groundset']
if not isinstance(kwds['circuit_closures'], dict):
# Convert to dictionary
CC = {}
for X in kwds['circuit_closures']:
if X[0] not in CC:
CC[X[0]] = []
CC[X[0]].append(X[1])
else:
CC = kwds['circuit_closures']
M = CircuitClosuresMatroid(groundset=E, circuit_closures=CC)
# Matroids:
if 'matroid' in kwds:
M = kwds['matroid']
if not isinstance(M, sage.matroids.matroid.Matroid):
raise ValueError("input does not appear to be of Matroid type.")
# Regular option:
if want_regular:
M = sage.matroids.utilities.make_regular_matroid_from_matroid(M)
if want_check:
if not M.is_valid():
raise ValueError(
'input does not correspond to a valid regular matroid.')
return M
class Isogeny_graph:
# Class for construction isogeny graphs with 1 or more degrees
# ls is a list of primes which define the defining degrees
def __init__(self, E, ls, special=False):
self._ls = ls
j = E.j_invariant()
self.field = E.base_field()
self._graph = Graph(multiedges=True, loops=True)
self._special = False
self._graph.add_vertex(j)
queue = [j]
R = rainbow(len(ls) + 1)
self._rainbow = R
self._edge_colors = {R[i]: [] for i in range(len(ls))}
while queue:
color = 0
s = queue.pop(0)
if s == 0 or s == 1728:
self._special = True
for l in ls:
neighb = isogenous_curves(s, l)
for i in neighb:
if not self._graph.has_vertex(i[0]):
queue.append(i[0])
self._graph.add_vertex(i[0])
if not ((s, i[0], l) in self._edge_colors[R[color]] or
(i[0], s, l) in self._edge_colors[R[color]]):
for _ in range(i[1]):
self._graph.add_edge((s, i[0], l))
self._edge_colors[R[color]].append((s, i[0], l))
color += 1
if self._special and special:
print("Curve with j_invariant 0 or 1728 found, may malfunction.")
def __repr__(self):
return "Isogeny graph of degrees %r" % (self._ls)
# Returns degrees of isogenies
def degrees(self):
return self._ls
# Returns list of all edges
def edges(self):
return self._graph.edges()
# Returns list of all vertices
def vertices(self):
return self._graph.vertices()
# Plots the graph
# Optional arguments figsize, vertex_size, vertex_labels and layout which are the same as in igraph
def plot(self,
figsize=None,
edge_labels=False,
vertex_size=None,
layout=None):
if vertex_size == None:
return self._graph.plot(edge_colors=self._edge_colors,
figsize=figsize,
edge_labels=edge_labels,
layout=layout)
else:
return self._graph.plot(edge_colors=self._edge_colors,
figsize=figsize,
edge_labels=edge_labels,
vertex_size=vertex_size,
layout=layout)
class Volcano:
# Class for l-volcano of elliptic curve E over finite field
# We can construct Volcano either using Velu algorithm (Velu = True) or modular polynomials (Velu = False)
# If Velu algorithm is chosen, constructor tries to find kernels of isogenies in extension,
# if the size of the base field of extension is higher than upper_bit_limit, then the algorithm stops
# In case of the option of modular polynomials, we assume that l<127
# You can turn off the 0,1728 warning with special = False
def __init__(self, E, l, Velu=False, upper_bit_limit=100, special=True):
self._l = l
self._E = E
global VELU
VELU = Velu
global UPPER_BITS
UPPER_BITS = upper_bit_limit
self.field = E.base_field()
try:
self._neighbours, self._vertices, self._special = BFS(
E.j_invariant(), l)
except large_extension_error as e:
print(
"Upper limit for bitlength of size of extension field exceeded"
)
print(e.msg)
return
if self._special and special:
print("Curve with j_invariant 0 or 1728 found, may malfunction.")
self._depths = {}
self._levels = []
self._depth = 0
self._graph = Graph(multiedges=True, loops=True)
for s in self._neighbours.values():
self._graph.add_vertex(s[0])
for s in self._neighbours.values():
for i in range(1, len(s)):
if not self._graph.has_edge(s[0], s[i][0]):
for j in range(s[i][1]):
self._graph.add_edge((s[0], s[i][0], j))
if len(self._vertices) > 1 and E.is_ordinary():
self.compute_depths()
else:
self._depths = {str(E.j_invariant()): 0}
self._levels = [self._vertices]
# Method for computing depths and sorting vertices into levels
def compute_depths(self):
heights = {}
level = []
if len(list(self._neighbours.values())[0]) == 3:
self._levels = [self._vertices]
self._depths = {str(i): 0 for i in self._vertices}
self._depth = 0
return
for s in self._neighbours.keys():
if len(self._neighbours[s]) == 2:
heights[s] = 0
level.append(self._neighbours[s][0])
self._levels.append(level)
h = 1
while len(heights.keys()) != len(self._vertices):
level = []
for s in self._neighbours.keys():
if s in heights.keys():
continue
if len(self._neighbours[s]) > 2:
if str(self._neighbours[s][1][0]) in heights.keys(
) and heights[str(self._neighbours[s][1][0])] == h - 1:
heights[s] = h
level.append(self._neighbours[s][0])
continue
if str(self._neighbours[s][2][0]) in heights.keys(
) and heights[str(self._neighbours[s][2][0])] == h - 1:
heights[s] = h
level.append(self._neighbours[s][0])
continue
if len(self._neighbours[s]) > 3:
if str(self._neighbours[s][3][0]) in heights.keys(
) and heights[str(self._neighbours[s][3][0])] == h - 1:
heights[s] = h
level.append(self._neighbours[s][0])
continue
h += 1
self._levels.append(level)
self._depth = h - 1
self._depths = {}
for k in heights.keys():
self._depths[k] = h - 1 - heights[k]
self._levels.reverse()
# Returns the defining degree of volcano
def degree(self):
return self._l
# Returns the level at depth i
def level(self, i):
return self._levels[i]
# Returns list of all edges
def edges(self):
return self._graph.edges()
# Returns depth of volcano
def depth(self):
return self._depth
# Returns the crater of volcano
def crater(self):
return self._levels[0]
# Plots the volcano
# Optional arguments figsize, vertex_size, vertex_labels and layout which are the same as in igraph (sage Class)
def plot(self,
figsize=None,
vertex_labels=True,
vertex_size=None,
layout=None):
try:
self._graph.layout(layout=layout, save_pos=True)
except:
pass
if vertex_size != None:
return self._graph.plot(figsize=figsize,
vertex_labels=vertex_labels,
vertex_size=vertex_size)
else:
return self._graph.plot(figsize=figsize,
vertex_labels=vertex_labels)
# Returns all vertices of volcano
def vertices(self):
return self._vertices
# Returns all neighbours of vertex j
def neighbors(self, j):
return self._neighbours[str(j)][1:]
# Returns depth of vertex j
def vertex_depth(self, j):
return self._depths[str(j)]
# Returns true if the volcano contains vertex 1728 or 0
def special(self):
return self._special
# Returns parent (upper level neighbour) of j
def volcano_parent(self, j):
h = self._depths[str(j)]
for i in self._neighbours[str(j)][1:]:
if self._depths[str(i[0])] < h:
return i[0]
return None
# Returns all children (lower level neighbours) of vertex j
def volcano_children(self, j):
children = []
h = self._depths[str(j)]
for i in self._neighbours[str(j)][1:]:
if self._depths[str(i[0])] > h:
children.append(i[0])
return children
# Finds an extension of curve over which the volcano has depth h
def expand_volcano(self, h):
return Volcano(expand_volcano(self._E, h, self._l), self._l)
def __repr__(self):
return "Isogeny %r-volcano of depth %r over %r" % (
self._l, self.depth(), self.field)
def Matroid(*args, **kwds):
r"""
Construct a matroid.
Matroids are combinatorial structures that capture the abstract properties
of (linear/algebraic/...) dependence. Formally, a matroid is a pair
`M = (E, I)` of a finite set `E`, the *groundset*, and a collection of
subsets `I`, the independent sets, subject to the following axioms:
* `I` contains the empty set
* If `X` is a set in `I`, then each subset of `X` is in `I`
* If two subsets `X`, `Y` are in `I`, and `|X| > |Y|`, then there exists
`x \in X - Y` such that `Y + \{x\}` is in `I`.
See the :wikipedia:`Wikipedia article on matroids <Matroid>` for more
theory and examples. Matroids can be obtained from many types of
mathematical structures, and Sage supports a number of them.
There are two main entry points to Sage's matroid functionality. For
built-in matroids, do the following:
* Within a Sage session, type "matroids." (Do not press "Enter", and do
not forget the final period ".")
* Hit "tab".
You will see a list of methods which will construct matroids. For
example::
sage: F7 = matroids.named_matroids.Fano()
sage: len(F7.nonspanning_circuits())
7
or::
sage: U36 = matroids.Uniform(3, 6)
sage: U36.equals(U36.dual())
True
To define your own matroid, use the function ``Matroid()``.
This function attempts to interpret its arguments to create an appropriate
matroid. The following named arguments are supported:
INPUT:
- ``groundset`` -- If provided, the groundset of the matroid.
If not provided, the function attempts to determine a groundset from the
data.
- ``bases`` -- The list of bases (maximal independent sets) of the
matroid.
- ``independent_sets`` -- The list of independent sets of the matroid.
- ``circuits`` -- The list of circuits of the matroid.
- ``graph`` -- A graph, whose edges form the elements of the matroid.
- ``matrix`` -- A matrix representation of the matroid.
- ``reduced_matrix`` -- A reduced representation of the matroid: if
``reduced_matrix = A``
then the matroid is represented by `[I\ \ A]` where `I` is an
appropriately sized identity matrix.
- ``rank_function`` -- A function that computes the rank of each subset.
Can only be provided together with a groundset.
- ``circuit_closures`` -- Either a list of tuples ``(k, C)`` with ``C``
the closure of a circuit, and ``k`` the rank of ``C``, or a dictionary
``D`` with ``D[k]`` the set of closures of rank-``k`` circuits.
- ``matroid`` -- An object that is already a matroid. Useful only with the
``regular`` option.
Up to two unnamed arguments are allowed.
- One unnamed argument, no named arguments other than ``regular`` -- the
input should be either a graph, or a matrix, or a list of independent
sets containing all bases, or a matroid.
- Two unnamed arguments: the first is the groundset, the second a graph,
or a matrix, or a list of independent sets containing all bases, or a
matroid.
- One unnamed argument, at least one named argument: the unnamed argument
is the groundset, the named argument is as above (but must be different
from ``groundset``).
The examples section details how each of the input types deals with
explicit or implicit groundset arguments.
OPTIONS:
- ``regular`` -- (default: ``False``) boolean. If ``True``,
output a
:class:`RegularMatroid <sage.matroids.linear_matroid.RegularMatroid>`
instance such that, *if* the input defines a valid regular matroid, then
the output represents this matroid. Note that this option can be
combined with any type of input.
- ``ring`` -- any ring. If provided, and the input is a ``matrix`` or
``reduced_matrix``, output will be a linear matroid over the ring or
field ``ring``.
- ``field`` -- any field. Same as ``ring``, but only fields are allowed.
- ``check`` -- (default: ``True``) boolean. If ``True`` and
``regular`` is true, the output is checked to make sure it is a valid
regular matroid.
.. WARNING::
Except for regular matroids, the input is not checked for validity. If
your data does not correspond to an actual matroid, the behavior of
the methods is undefined and may cause strange errors. To ensure you
have a matroid, run
:meth:`M.is_valid() <sage.matroids.matroid.Matroid.is_valid>`.
.. NOTE::
The ``Matroid()`` method will return instances of type
:class:`BasisMatroid <sage.matroids.basis_matroid.BasisMatroid>`,
:class:`CircuitClosuresMatroid <sage.matroids.circuit_closures_matroid.CircuitClosuresMatroid>`,
:class:`LinearMatroid <sage.matroids.linear_matroid.LinearMatroid>`,
:class:`BinaryMatroid <sage.matroids.linear_matroid.LinearMatroid>`,
:class:`TernaryMatroid <sage.matroids.linear_matroid.LinearMatroid>`,
:class:`QuaternaryMatroid <sage.matroids.linear_matroid.LinearMatroid>`,
:class:`RegularMatroid <sage.matroids.linear_matroid.LinearMatroid>`, or
:class:`RankMatroid <sage.matroids.rank_matroid.RankMatroid>`. To
import these classes (and other useful functions) directly into Sage's
main namespace, type::
sage: from sage.matroids.advanced import *
See :mod:`sage.matroids.advanced <sage.matroids.advanced>`.
EXAMPLES:
Note that in these examples we will often use the fact that strings are
iterable in these examples. So we type ``'abcd'`` to denote the list
``['a', 'b', 'c', 'd']``.
#. List of bases:
All of the following inputs are allowed, and equivalent::
sage: M1 = Matroid(groundset='abcd', bases=['ab', 'ac', 'ad',
....: 'bc', 'bd', 'cd'])
sage: M2 = Matroid(bases=['ab', 'ac', 'ad', 'bc', 'bd', 'cd'])
sage: M3 = Matroid(['ab', 'ac', 'ad', 'bc', 'bd', 'cd'])
sage: M4 = Matroid('abcd', ['ab', 'ac', 'ad', 'bc', 'bd', 'cd'])
sage: M5 = Matroid('abcd', bases=[['a', 'b'], ['a', 'c'],
....: ['a', 'd'], ['b', 'c'],
....: ['b', 'd'], ['c', 'd']])
sage: M1 == M2
True
sage: M1 == M3
True
sage: M1 == M4
True
sage: M1 == M5
True
We do not check if the provided input forms an actual matroid::
sage: M1 = Matroid(groundset='abcd', bases=['ab', 'cd'])
sage: M1.full_rank()
2
sage: M1.is_valid()
False
Bases may be repeated::
sage: M1 = Matroid(['ab', 'ac'])
sage: M2 = Matroid(['ab', 'ac', 'ab'])
sage: M1 == M2
True
#. List of independent sets:
::
sage: M1 = Matroid(groundset='abcd',
....: independent_sets=['', 'a', 'b', 'c', 'd', 'ab',
....: 'ac', 'ad', 'bc', 'bd', 'cd'])
We only require that the list of independent sets contains each basis
of the matroid; omissions of smaller independent sets and
repetitions are allowed::
sage: M1 = Matroid(bases=['ab', 'ac'])
sage: M2 = Matroid(independent_sets=['a', 'ab', 'b', 'ab', 'a',
....: 'b', 'ac'])
sage: M1 == M2
True
#. List of circuits:
::
sage: M1 = Matroid(groundset='abc', circuits=['bc'])
sage: M2 = Matroid(bases=['ab', 'ac'])
sage: M1 == M2
True
A matroid specified by a list of circuits gets converted to a
:class:`BasisMatroid <sage.matroids.basis_matroid.BasisMatroid>`
internally::
sage: M = Matroid(groundset='abcd', circuits=['abc', 'abd', 'acd',
....: 'bcd'])
sage: type(M)
<type 'sage.matroids.basis_matroid.BasisMatroid'>
Strange things can happen if the input does not satisfy the circuit
axioms, and these are not always caught by the
:meth:`is_valid() <sage.matroids.matroid.Matroid.is_valid>` method. So
always check whether your input makes sense!
::
sage: M = Matroid('abcd', circuits=['ab', 'acd'])
sage: M.is_valid()
True
sage: [sorted(C) for C in M.circuits()]
[['a']]
#. Graph:
Sage has great support for graphs, see :mod:`sage.graphs.graph`.
::
sage: G = graphs.PetersenGraph()
sage: Matroid(G)
Regular matroid of rank 9 on 15 elements with 2000 bases
Note: if a groundset is specified, we assume it is in the same order
as
:meth:`G.edge_iterator() <sage.graphs.generic_graph.GenericGraph.edge_iterator>`
provides::
sage: G = Graph([(0, 1), (0, 2), (0, 2), (1, 2)])
sage: M = Matroid('abcd', G)
sage: M.rank(['b', 'c'])
1
If no groundset is provided, we attempt to use the edge labels::
sage: G = Graph([(0, 1, 'a'), (0, 2, 'b'), (1, 2, 'c')])
sage: M = Matroid(G)
sage: sorted(M.groundset())
['a', 'b', 'c']
If no edge labels are present and the graph is simple, we use the
tuples ``(i, j)`` of endpoints. If that fails, we simply use a list
``[0..m-1]`` ::
sage: G = Graph([(0, 1), (0, 2), (1, 2)])
sage: M = Matroid(G)
sage: sorted(M.groundset())
[(0, 1), (0, 2), (1, 2)]
sage: G = Graph([(0, 1), (0, 2), (0, 2), (1, 2)])
sage: M = Matroid(G)
sage: sorted(M.groundset())
[0, 1, 2, 3]
When the ``graph`` keyword is used, a variety of inputs can be
converted to a graph automatically. The following uses a graph6 string
(see the :class:`Graph <sage.graphs.graph.Graph>` method's
documentation)::
sage: Matroid(graph=':I`AKGsaOs`cI]Gb~')
Regular matroid of rank 9 on 17 elements with 4004 bases
However, this method is no more clever than ``Graph()``::
sage: Matroid(graph=41/2)
Traceback (most recent call last):
...
ValueError: input does not seem to represent a graph.
#. Matrix:
The basic input is a
:mod:`Sage matrix <sage.matrix.constructor>`::
sage: A = Matrix(GF(2), [[1, 0, 0, 1, 1, 0],
....: [0, 1, 0, 1, 0, 1],
....: [0, 0, 1, 0, 1, 1]])
sage: M = Matroid(matrix=A)
sage: M.is_isomorphic(matroids.CompleteGraphic(4))
True
Various shortcuts are possible::
sage: M1 = Matroid(matrix=[[1, 0, 0, 1, 1, 0],
....: [0, 1, 0, 1, 0, 1],
....: [0, 0, 1, 0, 1, 1]], ring=GF(2))
sage: M2 = Matroid(reduced_matrix=[[1, 1, 0],
....: [1, 0, 1],
....: [0, 1, 1]], ring=GF(2))
sage: M3 = Matroid(groundset=[0, 1, 2, 3, 4, 5],
....: matrix=[[1, 1, 0], [1, 0, 1], [0, 1, 1]],
....: ring=GF(2))
sage: A = Matrix(GF(2), [[1, 1, 0], [1, 0, 1], [0, 1, 1]])
sage: M4 = Matroid([0, 1, 2, 3, 4, 5], A)
sage: M1 == M2
True
sage: M1 == M3
True
sage: M1 == M4
True
However, with unnamed arguments the input has to be a ``Matrix``
instance, or the function will try to interpret it as a set of bases::
sage: Matroid([0, 1, 2], [[1, 0, 1], [0, 1, 1]])
Traceback (most recent call last):
...
ValueError: basis has wrong cardinality.
If the groundset size equals number of rows plus number of columns, an
identity matrix is prepended. Otherwise the groundset size must equal
the number of columns::
sage: A = Matrix(GF(2), [[1, 1, 0], [1, 0, 1], [0, 1, 1]])
sage: M = Matroid([0, 1, 2], A)
sage: N = Matroid([0, 1, 2, 3, 4, 5], A)
sage: M.rank()
2
sage: N.rank()
3
We automatically create an optimized subclass, if available::
sage: Matroid([0, 1, 2, 3, 4, 5],
....: matrix=[[1, 1, 0], [1, 0, 1], [0, 1, 1]],
....: field=GF(2))
Binary matroid of rank 3 on 6 elements, type (2, 7)
sage: Matroid([0, 1, 2, 3, 4, 5],
....: matrix=[[1, 1, 0], [1, 0, 1], [0, 1, 1]],
....: field=GF(3))
Ternary matroid of rank 3 on 6 elements, type 0-
sage: Matroid([0, 1, 2, 3, 4, 5],
....: matrix=[[1, 1, 0], [1, 0, 1], [0, 1, 1]],
....: field=GF(4, 'x'))
Quaternary matroid of rank 3 on 6 elements
sage: Matroid([0, 1, 2, 3, 4, 5],
....: matrix=[[1, 1, 0], [1, 0, 1], [0, 1, 1]],
....: field=GF(2), regular=True)
Regular matroid of rank 3 on 6 elements with 16 bases
Otherwise the generic LinearMatroid class is used::
sage: Matroid([0, 1, 2, 3, 4, 5],
....: matrix=[[1, 1, 0], [1, 0, 1], [0, 1, 1]],
....: field=GF(83))
Linear matroid of rank 3 on 6 elements represented over the Finite
Field of size 83
An integer matrix is automatically converted to a matrix over `\QQ`.
If you really want integers, you can specify the ring explicitly::
sage: A = Matrix([[1, 1, 0], [1, 0, 1], [0, 1, -1]])
sage: A.base_ring()
Integer Ring
sage: M = Matroid([0, 1, 2, 3, 4, 5], A)
sage: M.base_ring()
Rational Field
sage: M = Matroid([0, 1, 2, 3, 4, 5], A, ring=ZZ)
sage: M.base_ring()
Integer Ring
#. Rank function:
Any function mapping subsets to integers can be used as input::
sage: def f(X):
....: return min(len(X), 2)
....:
sage: M = Matroid('abcd', rank_function=f)
sage: M
Matroid of rank 2 on 4 elements
sage: M.is_isomorphic(matroids.Uniform(2, 4))
True
#. Circuit closures:
This is often a really concise way to specify a matroid. The usual way
is a dictionary of lists::
sage: M = Matroid(circuit_closures={3: ['edfg', 'acdg', 'bcfg',
....: 'cefh', 'afgh', 'abce', 'abdf', 'begh', 'bcdh', 'adeh'],
....: 4: ['abcdefgh']})
sage: M.equals(matroids.named_matroids.P8())
True
You can also input tuples `(k, X)` where `X` is the closure of a
circuit, and `k` the rank of `X`::
sage: M = Matroid(circuit_closures=[(2, 'abd'), (3, 'abcdef'),
....: (2, 'bce')])
sage: M.equals(matroids.named_matroids.Q6())
True
#. Matroid:
Most of the time, the matroid itself is returned::
sage: M = matroids.named_matroids.Fano()
sage: N = Matroid(M)
sage: N is M
True
But it can be useful with the ``regular`` option::
sage: M = Matroid(circuit_closures={2:['adb', 'bec', 'cfa',
....: 'def'], 3:['abcdef']})
sage: N = Matroid(M, regular=True)
sage: N
Regular matroid of rank 3 on 6 elements with 16 bases
sage: Matrix(N)
[1 0 0 1 1 0]
[0 1 0 1 1 1]
[0 0 1 0 1 1]
The ``regular`` option:
::
sage: M = Matroid(reduced_matrix=[[1, 1, 0],
....: [1, 0, 1],
....: [0, 1, 1]], regular=True)
sage: M
Regular matroid of rank 3 on 6 elements with 16 bases
sage: M.is_isomorphic(matroids.CompleteGraphic(4))
True
By default we check if the resulting matroid is actually regular. To
increase speed, this check can be skipped::
sage: M = matroids.named_matroids.Fano()
sage: N = Matroid(M, regular=True)
Traceback (most recent call last):
...
ValueError: input does not correspond to a valid regular matroid.
sage: N = Matroid(M, regular=True, check=False)
sage: N
Regular matroid of rank 3 on 7 elements with 32 bases
sage: N.is_valid()
False
Sometimes the output is regular, but represents a different matroid
from the one you intended::
sage: M = Matroid(Matrix(GF(3), [[1, 0, 1, 1], [0, 1, 1, 2]]))
sage: N = Matroid(Matrix(GF(3), [[1, 0, 1, 1], [0, 1, 1, 2]]),
....: regular=True)
sage: N.is_valid()
True
sage: N.is_isomorphic(M)
False
"""
# These are the valid arguments:
inputS = set(['groundset', 'bases', 'independent_sets', 'circuits', 'graph', 'matrix', 'reduced_matrix', 'rank_function', 'circuit_closures', 'matroid'])
# process options
if 'regular' in kwds:
want_regular = kwds['regular']
kwds.pop('regular')
else:
want_regular = False
if 'check' in kwds:
want_check = kwds['check']
kwds.pop('check')
else:
want_check = True
base_ring = None
have_field = False
if 'field' in kwds:
base_ring = kwds['field']
kwds.pop('field')
have_field = True
if 'ring' in kwds:
raise ValueError("only one of ring and field can be specified.")
try:
if not base_ring.is_field():
raise TypeError("specified ``field`` is not a field.")
except AttributeError:
raise TypeError("specified ``field`` is not a field.")
if 'ring' in kwds:
base_ring = kwds['ring']
kwds.pop('ring')
try:
if not base_ring.is_ring():
raise TypeError("specified ``ring`` is not a ring.")
except AttributeError:
raise TypeError("specified ``ring`` is not a ring.")
# Process unnamed arguments
if len(args) > 0:
if 'groundset' in kwds:
raise ValueError('when using unnamed arguments, groundset must be the first unnamed argument or be implicit.')
if len(args) > 2:
raise ValueError('at most two unnamed arguments are allowed.')
if len(args) == 2 or len(kwds) > 0:
# First argument should be the groundset
kwds['groundset'] = args[0]
# Check for unnamed data
dataindex = -1
if len(args) == 2:
dataindex = 1
if len(args) == 1 and len(kwds) == 0:
dataindex = 0
if dataindex > -1:
# One unnamed argument, no named arguments
if isinstance(args[dataindex], sage.graphs.graph.Graph):
kwds['graph'] = args[dataindex]
elif isinstance(args[dataindex], sage.matrix.matrix.Matrix):
kwds['matrix'] = args[dataindex]
elif isinstance(args[dataindex], sage.matroids.matroid.Matroid):
kwds['matroid'] = args[dataindex]
else:
kwds['independent_sets'] = args[dataindex]
# Check for multiple types of input
if len(set(kwds).difference(inputS)) > 0:
raise ValueError("unknown input argument")
if ('grondset' in kwds and len(kwds) != 2) or ('groundset' not in kwds and len(kwds) > 1):
raise ValueError("only one type of input may be specified.")
# Bases:
if 'bases' in kwds:
if 'groundset' not in kwds:
gs = set()
for B in kwds['bases']:
gs.update(B)
kwds['groundset'] = gs
M = BasisMatroid(groundset=kwds['groundset'], bases=kwds['bases'])
# Independent sets:
if 'independent_sets' in kwds:
# Convert to list of bases first
rk = -1
bases = []
for I in kwds['independent_sets']:
if len(I) == rk:
bases.append(I)
elif len(I) > rk:
bases = [I]
rk = len(I)
if 'groundset' not in kwds:
gs = set()
for B in bases:
gs.update(B)
kwds['groundset'] = gs
M = BasisMatroid(groundset=kwds['groundset'], bases=bases)
# Circuits:
if 'circuits' in kwds:
# Convert to list of bases first
# Determine groundset (note that this cannot detect coloops)
if 'groundset' not in kwds:
gs = set()
for C in kwds['circuits']:
gs.update(C)
kwds['groundset'] = gs
# determine the rank by computing a basis
B = set(kwds['groundset'])
for C in kwds['circuits']:
I = B.intersection(C)
if len(I) >= len(C):
B.discard(I.pop())
rk = len(B)
# Construct the basis matroid of appropriate rank. Note: slow!
BB = [frozenset(B) for B in combinations(kwds['groundset'], rk) if not any([frozenset(C).issubset(B) for C in kwds['circuits']])]
M = BasisMatroid(groundset=kwds['groundset'], bases=BB)
# Graphs:
if 'graph' in kwds:
# Construct the incidence matrix
# NOTE: we are not using Sage's built-in method because
# 1) we would need to fix the loops anyway
# 2) Sage will sort the columns, making it impossible to keep labels!
G = kwds['graph']
if not isinstance(G, sage.graphs.generic_graph.GenericGraph):
try:
G = Graph(G)
except (ValueError, TypeError, NetworkXError):
raise ValueError("input does not seem to represent a graph.")
V = G.vertices()
E = G.edges()
n = G.num_verts()
m = G.num_edges()
A = Matrix(ZZ, n, m, 0)
mm = 0
for i, j, k in G.edge_iterator():
A[V.index(i), mm] = -1
A[V.index(j), mm] += 1 # So loops get 0
mm += 1
# Decide on the groundset
if 'groundset' not in kwds:
# 1. Attempt to use edge labels.
sl = G.edge_labels()
if len(sl) == len(set(sl)):
kwds['groundset'] = sl
# 2. If simple, use vertex tuples
elif not G.has_multiple_edges():
kwds['groundset'] = [(i, j) for i, j, k in G.edge_iterator()]
else:
# 3. Use numbers
kwds['groundset'] = range(m)
M = RegularMatroid(matrix=A, groundset=kwds['groundset'])
want_regular = False # Save some time, since result is already regular
# Matrices:
if 'matrix' in kwds or 'reduced_matrix' in kwds:
if 'matrix' in kwds:
A = kwds['matrix']
if 'reduced_matrix' in kwds:
A = kwds['reduced_matrix']
# Fix the representation
if not isinstance(A, sage.matrix.matrix.Matrix):
try:
if base_ring is not None:
A = Matrix(base_ring, A)
else:
A = Matrix(A)
except ValueError:
raise ValueError("input does not seem to contain a matrix.")
# Fix the ring
if base_ring is not None:
if A.base_ring() != base_ring:
A = A.change_ring(base_ring)
elif A.base_ring() == ZZ and not want_regular: # Usually a rational matrix is intended, we presume.
A = A.change_ring(QQ)
base_ring = QQ
else:
base_ring = A.base_ring()
# Determine groundset:
if 'matrix' in kwds:
if 'groundset' in kwds:
if len(kwds['groundset']) == A.nrows() + A.ncols():
kwds['reduced_matrix'] = A
kwds.pop('matrix')
else:
if len(kwds['groundset']) != A.ncols():
raise ValueError("groundset size does not correspond to matrix size.")
else:
kwds['groundset'] = range(A.ncols())
if 'reduced_matrix' in kwds:
if 'groundset' in kwds:
if len(kwds['groundset']) != A.nrows() + A.ncols():
raise ValueError("groundset size does not correspond to matrix size.")
else:
kwds['groundset'] = range(A.nrows() + A.ncols())
if 'matrix' in kwds:
if base_ring == GF(2):
M = BinaryMatroid(groundset=kwds['groundset'], matrix=A)
elif base_ring == GF(3):
M = TernaryMatroid(groundset=kwds['groundset'], matrix=A)
elif base_ring.is_field() and base_ring.order() == 4: # GF(4) can have different generators.
M = QuaternaryMatroid(groundset=kwds['groundset'], matrix=A)
else:
M = LinearMatroid(groundset=kwds['groundset'], matrix=A, ring=base_ring)
if 'reduced_matrix' in kwds:
if A.base_ring() == GF(2):
M = BinaryMatroid(groundset=kwds['groundset'], reduced_matrix=A)
elif A.base_ring() == GF(3):
M = TernaryMatroid(groundset=kwds['groundset'], reduced_matrix=A)
elif A.base_ring().is_field() and A.base_ring().order() == 4: # GF(4) can have different generators.
M = QuaternaryMatroid(groundset=kwds['groundset'], reduced_matrix=A)
else:
M = LinearMatroid(groundset=kwds['groundset'], reduced_matrix=A, ring=base_ring)
# Rank functions:
if 'rank_function' in kwds:
if 'groundset' not in kwds:
raise ValueError('for rank functions, groundset needs to be specified.')
M = RankMatroid(groundset=kwds['groundset'], rank_function=kwds['rank_function'])
# Circuit closures:
if 'circuit_closures' in kwds:
if 'groundset' not in kwds:
E = set()
if isinstance(kwds['circuit_closures'], dict):
for X in kwds['circuit_closures'].itervalues():
for Y in X:
E.update(Y)
else:
for X in kwds['circuit_closures']:
E.update(X[1])
else:
E = kwds['groundset']
if not isinstance(kwds['circuit_closures'], dict):
# Convert to dictionary
CC = {}
for X in kwds['circuit_closures']:
if X[0] not in CC:
CC[X[0]] = []
CC[X[0]].append(X[1])
else:
CC = kwds['circuit_closures']
M = CircuitClosuresMatroid(groundset=E, circuit_closures=CC)
# Matroids:
if 'matroid' in kwds:
M = kwds['matroid']
if not isinstance(M, sage.matroids.matroid.Matroid):
raise ValueError("input does not appear to be of Matroid type.")
# Regular option:
if want_regular:
M = sage.matroids.utilities.make_regular_matroid_from_matroid(M)
if want_check:
if not M.is_valid():
raise ValueError('input does not correspond to a valid regular matroid.')
return M
