r"""
Independent sets

This module implements the :class:`IndependentSets` class which can be used to :

- List the independent sets (or cliques) of a graph
- Count them (which is obviously faster)
- Test whether a set of vertices is an independent set

It can also be restricted to focus on (inclusionwise) maximal independent
sets. See the documentation of :class:`IndependentSets` for actual examples.

Classes and methods
-------------------

"""

from sage.data_structures.binary_matrix cimport *
from sage.misc.cachefunc import cached_method
from sage.graphs.base.static_dense_graph cimport dense_graph_init


cdef inline int ismaximal(binary_matrix_t g, int n, bitset_t s):
    cdef int i
    for i in range(n):
        if (not bitset_in(s, i)) and bitset_are_disjoint(g.rows[i], s):
            return False

    return True

cdef class IndependentSets:
    r"""
    The set of independent sets of a graph.

    For more information on independent sets, see the
    :wikipedia:`Independent_set_(graph_theory)`.

    INPUT:

    - ``G`` -- a graph

    - ``maximal`` -- boolean (default: ``False``); whether to only consider
      (inclusionwise) maximal independent sets.

    - ``complement`` -- boolean (default: ``False``); whether to consider the
      graph's complement (i.e. cliques instead of independent sets).

    ALGORITHM:

    The enumeration of independent sets is done naively : given an independent
    set, this implementation considers all ways to add a new vertex to it
    (while keeping it an independent set), and then creates new independent
    sets from all those that were created this way.

    The implementation, however, is not recursive.

    .. NOTE::

        This implementation of the enumeration of *maximal* independent sets is
        not much faster than NetworkX', which is surprising as it is written in
        Cython. This being said, the algorithm from NetworkX appears to be
        slightly different from this one, and that would be a good thing to
        explore if one wants to improve the implementation.

        A simple generalization can also be done without too much modifications:
        iteration through independent sets with given size bounds
        (minimum and maximum number of vertices allowed).

    EXAMPLES:

    Listing all independent sets of the Claw graph::

        sage: from sage.graphs.independent_sets import IndependentSets
        sage: g = graphs.ClawGraph()
        sage: I = IndependentSets(g)
        sage: list(I)
        [[0], [1], [1, 2], [1, 2, 3], [1, 3], [2], [2, 3], [3], []]

    Count them::

        sage: I.cardinality()
        9

    List only the maximal independent sets::

        sage: Im = IndependentSets(g, maximal=True)
        sage: list(Im)
        [[0], [1, 2, 3]]

    And count them::

        sage: Im.cardinality()
        2

    One can easily count the number of independent sets of each cardinality::

        sage: g = graphs.PetersenGraph()
        sage: number_of = [0] * g.order()
        sage: for x in IndependentSets(g):
        ....:     number_of[len(x)] += 1
        sage: number_of
        [1, 10, 30, 30, 5, 0, 0, 0, 0, 0]

    It is also possible to define an iterator over all independent sets of a
    given cardinality. Note, however, that Sage will generate them *all*, to
    return only those that satisfy the cardinality constraints. Getting the list
    of independent sets of size 4 in this way can thus take a very long time::

        sage: is4 = (x for x in IndependentSets(g) if len(x) == 4)
        sage: list(is4)
        [[0, 2, 8, 9], [0, 3, 6, 7], [1, 3, 5, 9], [1, 4, 7, 8], [2, 4, 5, 6]]

    Given a subset of the vertices, it is possible to test whether it is an
    independent set::

        sage: g = graphs.DurerGraph()
        sage: I = IndependentSets(g)
        sage: [0, 2] in I
        True
        sage: [0, 3, 5] in I
        False

    If an element of the subset is not a vertex, then an error is raised::

        sage: [0, 'a', 'b', 'c'] in I
        Traceback (most recent call last):
        ...
        ValueError: a is not a vertex of the graph
    """
    def __init__(self, G, maximal=False, complement=False):
        r"""
        Constructor for this class

        TESTS::

            sage: from sage.graphs.independent_sets import IndependentSets
            sage: IndependentSets(graphs.PetersenGraph())
            <sage.graphs.independent_sets.IndependentSets object...

        Compute the number of matchings, and check with Sage's implementation::

            sage: from sage.graphs.independent_sets import IndependentSets
            sage: from sage.graphs.matchpoly import matching_polynomial
            sage: def check_matching(G):
            ....:     number_of_matchings = sum(map(abs, matching_polynomial(G).coefficients(sparse=False)))
            ....:     if number_of_matchings != IndependentSets(G.line_graph()).cardinality():
            ....:         raise ValueError("something goes wrong")
            sage: for i in range(30):
            ....:     check_matching(graphs.RandomGNP(11, .3))

        Compare the result with the output of :meth:`subgraph_search`::

            sage: from sage.sets.set import Set
            sage: def check_with_subgraph_search(G):
            ....:     IS = set(map(Set, list(IndependentSets(G))))
            ....:     if not all(G.subgraph(l).is_independent_set() for l in IS):
            ....:        print("Gloops")
            ....:     alpha = max(map(len, IS))
            ....:     IS2 = [Set([x]) for x in range(G.order())] + [Set()]
            ....:     for n in range(2, alpha + 1):
            ....:         IS2.extend(map(Set, list(G.subgraph_search_iterator(Graph(n), induced=True))))
            ....:     if len(IS) != len(set(IS2)):
            ....:        raise ValueError("something goes wrong")
            sage: for i in range(5):
            ....:     check_with_subgraph_search(graphs.RandomGNP(11, .3))

        Empty graph::

            sage: IS0 = IndependentSets(graphs.EmptyGraph())
            sage: list(IS0)
            [[]]
            sage: IS0.cardinality()
            1
        """
        cdef int i

        # Map from Vertex to Integer, and from Integer to Vertex
        self.vertices = list(G)
        self.vertex_to_int = {v: i for i, v in enumerate(self.vertices)}
        self.n = G.order()
        self.maximal = maximal
        dense_graph_init(self.g, G, translation=self.vertex_to_int)

        # If we must consider the graph's complement instead
        if complement:
            binary_matrix_complement(self.g)
            for i in range(self.n):
                binary_matrix_set0(self.g, i, i)

        self.count_only = 0

    def __iter__(self):
        r"""
        Return an iterator over the independent sets of ``self``.

        TESTS::

            sage: from sage.graphs.independent_sets import IndependentSets
            sage: I = IndependentSets(graphs.PetersenGraph())
            sage: iter1 = iter(I)
            sage: iter2 = iter(I)
            sage: next(iter1)      # indirect doctest
            [0]
            sage: next(iter2)      # indirect doctest
            [0]
            sage: next(iter2)
            [0, 2]
            sage: next(iter1)
            [0, 2]
        """
        if not self.n:
            yield []
            return

        cdef int i = 0

        cdef bitset_t current_set
        cdef bitset_t tmp
        bitset_init(current_set, self.n)
        bitset_set_first_n(current_set, 0)
        bitset_add(current_set, 0)
        bitset_init(tmp, self.n)

        cdef uint64_t count = 0
        cdef list ans
        cdef int j

        try:
            # At every moment of the algorithm current_set represents an independent
            # set, except for the ith bit. All bits >i are zero.

            while True:

                # If i is in current_set
                if bitset_in(current_set, i):

                    # We have found an independent set !
                    if bitset_are_disjoint(self.g.rows[i], current_set):

                        # Saving that set
                        bitset_copy(tmp, current_set)

                        # Preparing for the next set, except if we set the last bit.
                        if i < self.n - 1:

                            # Adding (i+1)th bit
                            bitset_add(current_set, i + 1)
                            i += 1
                        else:
                            bitset_discard(current_set, i)

                        # Returning the result if necessary ...
                        if self.maximal and not ismaximal(self.g, self.n, tmp):
                            continue

                        count += 1

                        if not self.count_only:
                            yield [self.vertices[j] for j in range(i + 1) if bitset_in(tmp,j)]
                            continue

                    else:
                        # Removing the ith bit
                        bitset_discard(current_set, i)

                        # Preparing for the next set !
                        if i < self.n - 1:
                            bitset_add(current_set, i + 1)
                            i += 1

                # Not already included in the set
                else:
                    if not i:
                        break

                    # Going backward, we explored all we could there !
                    if bitset_in(current_set, i - 1):
                        bitset_discard(current_set, i - 1)
                        bitset_add(current_set, i)
                    else:
                        i -= 1

            if not self.maximal:
                count += 1
                if not self.count_only:
                    yield []

            if self.count_only:
                yield count

        finally:
            bitset_free(current_set)
            bitset_free(tmp)

    def __dealloc__(self):
        r"""
        Frees everything we ever allocated
        """
        if self.g.rows:
            binary_matrix_free(self.g)

    @cached_method
    def cardinality(self):
        r"""
        Compute and return the number of independent sets.

        TESTS::

            sage: from sage.graphs.independent_sets import IndependentSets
            sage: IndependentSets(graphs.PetersenGraph()).cardinality()
            76

        Only maximal ones::

            sage: from sage.graphs.independent_sets import IndependentSets
            sage: IndependentSets(graphs.PetersenGraph(), maximal=True).cardinality()
            15
        """
        if not self.n:
            return 1

        self.count_only = 1

        for i in self:
            pass

        self.count_only = 0

        from sage.rings.integer import Integer
        return Integer(i)

    def __contains__(self, S):
        r"""
        Check whether the set is an independent set (possibly maximal)

        INPUT:

        - ``S`` -- a set of vertices to be tested.

        TESTS:

        All independent sets of PetersenGraph are... independent sets::

            sage: from sage.graphs.independent_sets import IndependentSets
            sage: G = graphs.PetersenGraph()
            sage: IS = IndependentSets(graphs.PetersenGraph())
            sage: all(s in IS for s in IS)
            True

        And only them are::

            sage: IS2 = [x for x in subsets(G) if x in IS]
            sage: sorted(IS) == sorted(IS2)
            True

        Same with maximal independent sets::

            sage: IS = IndependentSets(graphs.PetersenGraph(), maximal=True)
            sage: S = Subsets(G)
            sage: all(s in IS for s in IS)
            True
            sage: IS2 = [x for x in subsets(G) if x in IS]
            sage: sorted(IS) == sorted(IS2)
            True

        Check that the empty graph is dealt with correctly::

            sage: IS = IndependentSets(Graph())
            sage: [] in IS
            True
        """
        if not self.n:
            return not S

        cdef int i
        # Set of vertices as a bitset
        cdef bitset_t s
        bitset_init(s, self.n)

        try:
            bitset_set_first_n(s, 0)

            for I in S:
                try:
                    i = self.vertex_to_int[I]
                except KeyError:
                    raise ValueError(str(I) + " is not a vertex of the graph")

                # Adding the new vertex to s
                bitset_add(s, i)

                # Checking that the set s is independent
                if not bitset_are_disjoint(self.g.rows[i], s):
                    return False

            if self.maximal and not ismaximal(self.g, self.n, s):
                return False

            return True

        finally:
            bitset_free(s)
