r"""
Matroid construction

Theory
======

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. The object
:class:`matroids. <sage.matroids.matroids_catalog>` contains a number of
constructors for well-known matroids. The function
:func:`Matroid() <sage.matroids.constructor.Matroid>` allows you to define
your own matroids from a variety of sources. We briefly introduce both below;
follow the links for more comprehensive documentation.

Each matroid object in Sage comes with a number of built-in operations. An
overview can be found in the documentation of
:mod:`the abstract matroid class <sage.matroids.matroid>`.

Built-in matroids
=================

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: M = matroids.Wheel(4)
   sage: M.is_connected()
   True

or::

   sage: U36 = matroids.Uniform(3, 6)
   sage: U36.equals(U36.dual())
   True

A number of special matroids are collected under a ``named_matroids`` submenu.
To see which, type ``matroids.named_matroids.<tab>`` as above::

    sage: F7 = matroids.named_matroids.Fano()
    sage: len(F7.nonspanning_circuits())
    7

Constructing matroids
=====================

To define your own matroid, use the function
:func:`Matroid() <sage.matroids.constructor.Matroid>`. This function attempts
to interpret its arguments to create an appropriate matroid. The input
arguments are documented in detail
:func:`below <sage.matroids.constructor.Matroid>`.

EXAMPLES::

   sage: A = Matrix(GF(2), [[1, 0, 0, 0, 1, 1, 1],
   ....:                    [0, 1, 0, 1, 0, 1, 1],
   ....:                    [0, 0, 1, 1, 1, 0, 1]])
   sage: M = Matroid(A)
   sage: M.is_isomorphic(matroids.named_matroids.Fano())
   True

   sage: M = Matroid(graphs.PetersenGraph())
   sage: M.rank()
   9

AUTHORS:

- Rudi Pendavingh, Michael Welsh, Stefan van Zwam (2013-04-01): initial
  version

Functions
=========
"""

#*****************************************************************************
#       Copyright (C) 2013 Rudi Pendavingh <rudi.pendavingh@gmail.com>
#       Copyright (C) 2013 Michael Welsh <michael@welsh.co.nz>
#       Copyright (C) 2013 Stefan van Zwam <stefanvanzwam@gmail.com>
#
#
#  Distributed under the terms of the GNU General Public License (GPL)
#  as published by the Free Software Foundation; either version 2 of
#  the License, or (at your option) any later version.
#                  http://www.gnu.org/licenses/
#*****************************************************************************

from __future__ import absolute_import

from six import itervalues

from itertools import combinations
from sage.matrix.constructor import Matrix
from sage.graphs.all import Graph
from sage.structure.element import is_Matrix
from sage.rings.all import ZZ, QQ
from sage.rings.finite_rings.finite_field_base import FiniteField
import sage.matroids.matroid
import sage.matroids.basis_exchange_matroid
from .rank_matroid import RankMatroid
from .circuit_closures_matroid import CircuitClosuresMatroid
from .basis_matroid import BasisMatroid
from .linear_matroid import LinearMatroid, RegularMatroid, BinaryMatroid, TernaryMatroid, QuaternaryMatroid
from .graphic_matroid import GraphicMatroid
import sage.matroids.utilities


def Matroid(groundset=None, data=None, **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`` -- (optional) If provided, the groundset of the
      matroid. Otherwise, the function attempts to determine a groundset
      from the data.

    Exactly one of the following inputs must be given (where ``data``
    must be a positional argument and anything else must be a keyword
    argument):

    - ``data`` -- a graph or a matrix or a RevLex-Index string or a list
      of independent sets containing all bases or a matroid.
    - ``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.
    - ``revlex`` -- the encoding as a string of ``0`` and ``*`` symbols.
      Used by [MatroidDatabase]_ and explained in [MMIB2012]_.
    - ``matroid`` -- An object that is already a matroid. Useful only with the
      ``regular`` option.

    Further 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)
            <... '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)
            Graphic matroid of rank 9 on 15 elements

        If each edge has a unique label, then those are used as the ground set
        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 there are parallel edges, then integers are used for the ground set.
        If there are no edges in parallel, and is not a complete list of labels,
        or the labels are not unique, then vertex tuples are used::

            sage: G = Graph([(0, 1, 'a'), (0, 2, 'b'), (1, 2, 'b')])
            sage: M = Matroid(G)
            sage: sorted(M.groundset())
            [(0, 1), (0, 2), (1, 2)]
            sage: H = Graph([(0, 1, 'a'), (0, 2, 'b'), (1, 2, 'b'), (1, 2, 'c')], multiedges=True)
            sage: N = Matroid(H)
            sage: sorted(N.groundset())
            [0, 1, 2, 3]

        The GraphicMatroid object forces its graph to be connected. If a
        disconnected graph is used as input, it will connect the components.

            sage: G1 = graphs.CycleGraph(3); G2 = graphs.DiamondGraph()
            sage: G = G1.disjoint_union(G2)
            sage: M = Matroid(G)
            sage: M
            Graphic matroid of rank 5 on 8 elements
            sage: M.graph()
            Looped multi-graph on 6 vertices
            sage: M.graph().is_connected()
            True
            sage: M.is_connected()
            False


        If the keyword ``regular`` is set to ``True``, the output will instead
        be an instance of ``RegularMatroid``.

        ::

            sage: G = Graph([(0, 1), (0, 2), (1, 2)])
            sage: M = Matroid(G, regular=True); M
            Regular matroid of rank 2 on 3 elements with 3 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)], multiedges=True)
            sage: M = Matroid('abcd', G)
            sage: M.rank(['b', 'c'])
            1

        As before,
        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, regular=True)
            sage: sorted(M.groundset())
            [(0, 1), (0, 2), (1, 2)]

            sage: G = Graph([(0, 1), (0, 2), (0, 2), (1, 2)], multiedges=True)
            sage: M = Matroid(G, regular=True)
            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~')
            Graphic matroid of rank 9 on 17 elements

        However, this method is no more clever than ``Graph()``::

            sage: Matroid(graph=41/2)
            Traceback (most recent call last):
            ...
            ValueError: This input cannot be turned into 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

    #.  RevLex-Index:

        This requires the ``groundset`` to be given and also needs a
        additional keyword argument ``rank`` to specify the rank of the
        matroid::

            sage: M = Matroid("abcdef", "000000******0**", rank=4); M
            Matroid of rank 4 on 6 elements with 8 bases
            sage: list(M.bases())
            [frozenset({'a', 'b', 'd', 'f'}),
             frozenset({'a', 'c', 'd', 'f'}),
             frozenset({'b', 'c', 'd', 'f'}),
             frozenset({'a', 'b', 'e', 'f'}),
             frozenset({'a', 'c', 'e', 'f'}),
             frozenset({'b', 'c', 'e', 'f'}),
             frozenset({'b', 'd', 'e', 'f'}),
             frozenset({'c', 'd', 'e', 'f'})]

        Only the ``0`` symbols really matter, any symbol can be used
        instead of ``*``:

            sage: Matroid("abcdefg", revlex="0++++++++0++++0+++++0+--++----+--++", rank=4)
            Matroid of rank 4 on 7 elements with 31 bases

        It is checked that the input makes sense (but not that it
        defines a matroid)::

            sage: Matroid("abcdef", "000000******0**")
            Traceback (most recent call last):
            ...
            TypeError: for RevLex-Index, the rank needs to be specified
            sage: Matroid("abcdef", "000000******0**", rank=3)
            Traceback (most recent call last):
            ...
            ValueError: expected string of length 20 (6 choose 3), got 15
            sage: M = Matroid("abcdef", "*0000000000000*", rank=4); M
            Matroid of rank 4 on 6 elements with 2 bases
            sage: M.is_valid()
            False

    #.  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 is not 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

    TESTS::

        sage: Matroid()
        Traceback (most recent call last):
        ...
        TypeError: no input data given for Matroid()
        sage: Matroid("abc", bases=["abc"], foo="bar")
        Traceback (most recent call last):
        ...
        TypeError: Matroid() got an unexpected keyword argument 'foo'
        sage: Matroid(data=["x"], matrix=Matrix(1,1))
        Traceback (most recent call last):
        ...
        TypeError: Matroid() got an unexpected keyword argument 'matrix'
        sage: Matroid(bases=["x"], matrix=Matrix(1,1))
        Traceback (most recent call last):
        ...
        TypeError: Matroid() got an unexpected keyword argument 'matrix'
        sage: Matroid(Matrix(1,1), ring=ZZ, field=QQ)
        Traceback (most recent call last):
        ...
        TypeError: Matroid() got an unexpected keyword argument 'ring'
        sage: Matroid(rank_function=lambda X: len(X))
        Traceback (most recent call last):
        ...
        TypeError: for rank functions, the groundset needs to be specified
        sage: Matroid(matroid="rubbish")
        Traceback (most recent call last):
        ...
        TypeError: input 'rubbish' is not a matroid
    """
    # process options
    want_regular = kwds.pop('regular', False)
    check = kwds.pop('check', True)

    base_ring = None
    if 'field' in kwds:
        base_ring = kwds.pop('field')
        if check and not base_ring.is_field():
            raise TypeError("{} is not a field".format(base_ring))
    elif 'ring' in kwds:
        base_ring = kwds.pop('ring')
        if check and not base_ring.is_ring():
            raise TypeError("{} is not a ring".format(base_ring))

    # "key" is the kind of data we got
    key = None
    if data is None:
        for k in ['bases', 'independent_sets', 'circuits', 'graph',
                'matrix', 'reduced_matrix', 'rank_function', 'revlex',
                'circuit_closures', 'matroid']:
            if k in kwds:
                data = kwds.pop(k)
                key = k
                break
        else:
            # Assume that the single positional argument was actually
            # the data (instead of the groundset)
            data = groundset
            groundset = None

    if key is None:
        if isinstance(data, sage.graphs.graph.Graph):
            key = 'graph'
        elif is_Matrix(data):
            key = 'matrix'
        elif isinstance(data, sage.matroids.matroid.Matroid):
            key = 'matroid'
        elif isinstance(data, str):
            key = 'revlex'
        elif data is None:
            raise TypeError("no input data given for Matroid()")
        else:
            key = 'independent_sets'

    # Bases:
    if key == 'bases':
        if groundset is None:
            groundset = set()
            for B in data:
                groundset.update(B)
        M = BasisMatroid(groundset=groundset, bases=data)

    # Independent sets:
    elif key == 'independent_sets':
        # Convert to list of bases first
        rk = -1
        bases = []
        for I in data:
            if len(I) == rk:
                bases.append(I)
            elif len(I) > rk:
                bases = [I]
                rk = len(I)
        if groundset is None:
            groundset = set()
            for B in bases:
                groundset.update(B)
        M = BasisMatroid(groundset=groundset, bases=bases)

    # Circuits:
    elif key == 'circuits':
        # Convert to list of bases first
        # Determine groundset (note that this cannot detect coloops)
        if groundset is None:
            groundset = set()
            for C in data:
                groundset.update(C)
        # determine the rank by computing a basis element
        b = set(groundset)
        for C in data:
            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(groundset, rk)
              if not any(frozenset(C).issubset(B) for C in data)]
        M = BasisMatroid(groundset=groundset, bases=BB)

    # Graphs:

    elif key == 'graph':
        if isinstance(data, sage.graphs.generic_graph.GenericGraph):
            G = data
        else:
            G = Graph(data)
        # Decide on the groundset
        m = G.num_edges()
        if groundset is None:
            # 1. Attempt to use edge labels.
            sl = G.edge_labels()
            if len(sl) == len(set(sl)):
                groundset = sl
                # 2. If simple, use vertex tuples
            elif not G.has_multiple_edges():
                groundset = [(i, j) for i, j, k in G.edge_iterator()]
            else:
                # 3. Use numbers
                groundset = list(range(m))
        if want_regular:
        # 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!
            V = G.vertices()
            n = G.num_verts()
            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
            M = RegularMatroid(matrix=A, groundset=groundset)
            want_regular = False  # Save some time, since result is already regular
        else:
            M = GraphicMatroid(G, groundset=groundset)

    # Matrices:
    elif key in ['matrix', 'reduced_matrix']:
        A = data
        is_reduced = (key == 'reduced_matrix')

        # Fix the representation
        if not is_Matrix(A):
            if base_ring is not None:
                A = Matrix(base_ring, A)
            else:
                A = Matrix(A)

        # Fix the ring
        if base_ring is not None:
            if A.base_ring() is not base_ring:
                A = A.change_ring(base_ring)
        elif A.base_ring() is 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()

        # Check groundset
        if groundset is not None:
            if not is_reduced:
                if len(groundset) == A.ncols():
                    pass
                elif len(groundset) == A.nrows() + A.ncols():
                    is_reduced = True
                else:
                    raise ValueError("groundset size does not correspond to matrix size")
            elif is_reduced:
                if len(groundset) == A.nrows() + A.ncols():
                    pass
                else:
                    raise ValueError("groundset size does not correspond to matrix size")

        if is_reduced:
            kw = dict(groundset=groundset, reduced_matrix=A)
        else:
            kw = dict(groundset=groundset, matrix=A)

        if isinstance(base_ring, FiniteField):
            q = base_ring.order()
        else:
            q = 0

        if q == 2:
            M = BinaryMatroid(**kw)
        elif q == 3:
            M = TernaryMatroid(**kw)
        elif q == 4:
            M = QuaternaryMatroid(**kw)
        else:
            M = LinearMatroid(ring=base_ring, **kw)

    # Rank functions:
    elif key == 'rank_function':
        if groundset is None:
            raise TypeError('for rank functions, the groundset needs to be specified')
        M = RankMatroid(groundset=groundset, rank_function=data)

    # RevLex-Index:
    elif key == "revlex":
        if groundset is None:
            raise TypeError('for RevLex-Index, the groundset needs to be specified')
        try:
            rk = kwds.pop("rank")
        except KeyError:
            raise TypeError('for RevLex-Index, the rank needs to be specified')

        groundset = tuple(groundset)
        data = tuple(data)
        rk = int(rk)
        N = len(groundset)

        def revlex_sort_key(s):
            return tuple(reversed(s))
        subsets = sorted(combinations(range(N), rk), key=revlex_sort_key)
        if len(data) != len(subsets):
            raise ValueError("expected string of length %s (%s choose %s), got %s" %
                (len(subsets), N, rk, len(data)))
        bases = []
        for i, x in enumerate(data):
            if x != '0':
                bases.append([groundset[c] for c in subsets[i]])
        M = BasisMatroid(groundset=groundset, bases=bases)

    # Circuit closures:
    elif key == 'circuit_closures':
        if isinstance(data, dict):
            CC = data
        else:
            # Convert to dictionary
            CC = {}
            for X in data:
                if X[0] not in CC:
                    CC[X[0]] = []
                CC[X[0]].append(X[1])

        if groundset is None:
            groundset = set()
            for X in itervalues(CC):
                for Y in X:
                    groundset.update(Y)

        M = CircuitClosuresMatroid(groundset=groundset, circuit_closures=CC)

    # Matroids:
    elif key == 'matroid':
        if not isinstance(data, sage.matroids.matroid.Matroid):
            raise TypeError("input {!r} is not a matroid".format(data))
        M = data

    else:
        raise AssertionError("unknown key %r" % key)

    # All keywords should be used
    for k in kwds:
        raise TypeError("Matroid() got an unexpected keyword argument '{}'".format(k))

    if want_regular:
        M = sage.matroids.utilities.make_regular_matroid_from_matroid(M)
        if check and not M.is_valid():
            raise ValueError('input is not a valid regular matroid')

    return M
