r"""
Covering designs: coverings of `t`-element subsets of a `v`-set by `k`-sets

A `(v, k, t)` covering design `C` is an incidence structure consisting of a
set of points `P` of order `v`, and a set of blocks `B`, where each
block contains `k` points of `P`.  Every `t`-element subset of `P`
must be contained in at least one block.

If every `t`-set is contained in exactly one block of `C`, then we
have a block design.  Following the block design implementation, the
standard representation of a covering design uses `P = [0, 1, ..., v-1]`.

In addition to the parameters and incidence structure for a covering
design from this database, we include extra information:

* Best known lower bound on the size of a `(v, k, t)`-covering design
* Name of the person(s) who produced the design
* Method of construction used
* Date when the design was added to the database

REFERENCES:

.. [1] La Jolla Covering Repository,
  https://ljcr.dmgordon.org/cover.html

.. [2] Daniel M. Gordon and Douglas R. Stinson, *Coverings*,
  Chapter 1 in: Charles J. Colbourn and Jeffrey H. Dinitz,
  *Handbook of Combinatorial Designs*, 2nd Edition, 2006.
  https://www.dmgordon.org/papers/hcd.pdf

AUTHORS:

- Daniel M. Gordon (2008-12-22): initial version

Classes and methods
-------------------
"""
# ****************************************************************************
#     Copyright (C) 2008 Daniel M. Gordon <dmgordo@gmail.com>
#
# Distributed under the terms of the GNU General Public License (GPL)
#
# The full text of the GPL is available at:
#
#                         https://www.gnu.org/licenses/
# ****************************************************************************

from urllib.request import urlopen

from sage.misc.sage_eval import sage_eval
from sage.structure.sage_object import SageObject
from sage.rings.rational import Rational
from sage.arith.all import binomial
from sage.combinat.combination import Combinations
from sage.combinat.designs.incidence_structures import IncidenceStructure
from sage.cpython.string import bytes_to_str


def schonheim(v, k, t):
    r"""
    Return the Schonheim lower bound for the size of such a covering design.

    INPUT:

    - ``v`` -- integer, size of point set

    - ``k`` -- integer, cardinality of each block

    - ``t`` -- integer, cardinality of sets being covered

    OUTPUT:

    The Schonheim lower bound for such a covering design's size:
    `C(v, k, t) \leq \lceil(\frac{v}{k} \lceil \frac{v-1}{k-1} \cdots
    \lceil \frac{v-t+1}{k-t+1} \rceil \cdots \rceil \rceil`

    EXAMPLES::

        sage: from sage.combinat.designs.covering_design import schonheim
        sage: schonheim(10, 3, 2)
        17
        sage: schonheim(32, 16, 8)
        930
    """
    bound = 1
    for i in range(t - 1, -1, -1):
        bound = Rational((bound * (v - i), k - i)).ceil()
    return bound


def trivial_covering_design(v, k, t):
    r"""
    Construct a trivial covering design.

    INPUT:

    - ``v`` -- integer, size of point set

    - ``k`` -- integer, cardinality of each block

    - ``t`` -- integer, cardinality of sets being covered

    OUTPUT:

    A trivial `(v, k, t)` covering design

    EXAMPLES::

        sage: C = trivial_covering_design(8, 3, 1)
        sage: print(C)
        C(8, 3, 1) = 3
        Method: Trivial
        0   1   2
        0   6   7
        3   4   5
        sage: C = trivial_covering_design(5, 3, 2)
        sage: print(C)
        4 <= C(5, 3, 2) <= 10
        Method: Trivial
        0   1   2
        0   1   3
        0   1   4
        0   2   3
        0   2   4
        0   3   4
        1   2   3
        1   2   4
        1   3   4
        2   3   4

    .. NOTE::

        Cases are:

        * `t=0`: This could be empty, but it's a useful convention to have
          one block (which is empty if $k=0$).

        * `t=1` : This contains `\lceil v/k \rceil` blocks:
          `[0, ..., k-1], [k, ..., 2k-1], ...`.  The last block wraps around if
          `k` does not divide `v`.

        * anything else: Just use every `k`-subset of `[0, 1,..., v-1]`.

    """
    if t == 0:  # single block [0, ..., k-1]
        blk = []
        for i in range(k):
            blk.append(i)
        return CoveringDesign(v, k, t, 1, range(v), [blk], 1, "Trivial")
    if t == 1:  # blocks [0, ..., k-1], [k, ..., 2k-1], ...
        size = Rational((v, k)).ceil()
        blocks = []
        for i in range(size - 1):
            blk = []
            for j in range(i * k, (i + 1) * k):
                blk.append(j)
            blocks.append(blk)
        blk = []  # last block: if k does not divide v, wrap around
        for j in range((size - 1) * k, v):
            blk.append(j)
        for j in range(k - len(blk)):
            blk.append(j)
        blk.sort()
        blocks.append(blk)
        return CoveringDesign(v, k, t, size, range(v), blocks, size, "Trivial")
    # default case, all k-subsets
    return CoveringDesign(v, k, t, binomial(v, k), range(v),
                          Combinations(range(v), k), schonheim(v, k, t),
                          "Trivial")


class CoveringDesign(SageObject):
    """
    Covering design.

    INPUT:

    - ``v``, ``k``, ``t`` -- integer parameters of the covering design

    - ``size`` (integer)

    - ``points`` -- list of points (default points are `[0, ..., v-1]`)

    - ``blocks``

    - ``low_bd`` (integer) -- lower bound for such a design

    - ``method``, ``creator``, ``timestamp`` -- database information
    """

    def __init__(self, v=0, k=0, t=0, size=0, points=None, blocks=None,
                 low_bd=0, method='', creator='', timestamp=''):
        """
        EXAMPLES::

            sage: C = CoveringDesign(5, 4, 3, 4, range(5), [[0, 1, 2, 3],
            ....:     [0, 1, 2, 4], [0, 1, 3, 4], [0, 2, 3, 4]], 4,
            ....:     'Lexicographic Covering')
            sage: print(C)
            C(5, 4, 3) = 4
            Method: Lexicographic Covering
            0   1   2   3
            0   1   2   4
            0   1   3   4
            0   2   3   4
        """
        self.__v = v
        self.__k = k
        self.__t = t
        self.__size = size
        if low_bd > 0:
            self.__low_bd = low_bd
        else:
            self.__low_bd = schonheim(v, k, t)
        self.__method = method
        self.__creator = creator
        self.__timestamp = timestamp
        if points is None:
            points = []
        if blocks is None:
            blocks = []
        self.__incidence_structure = IncidenceStructure(points, blocks)

    def __repr__(self):
        """
        Return the representation of this covering design.

        This has the parameters but not the blocks.

        EXAMPLES::

            sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
            ....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
            ....:     [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
            sage: C
            (7, 3, 2)-covering design of size 7
            Lower bound: 7
            Method: Projective Plane
        """
        repr = ('(%d, %d, %d)-covering design of size %d\n'
                % (self.__v, self.__k, self.__t, self.__size))
        repr += 'Lower bound: %d\n' % (self.__low_bd)
        if self.__creator:
            repr += 'Created by: %s\n' % (self.__creator)
        if self.__method:
            repr += 'Method: %s\n' % (self.__method)
        if self.__timestamp:
            repr += 'Submitted on: %s\n' % (self.__timestamp)
        return repr[:-1]

    def __str__(self):
        """
        Return the string for this covering design.

        This has the parameters and the blocks in readable form.

        EXAMPLES::

            sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
            ....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
            ....:     [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
            sage: print(C)
            C(7, 3, 2) = 7
            Method: Projective Plane
            0   1   2
            0   3   4
            0   5   6
            1   3   5
            1   4   6
            2   3   6
            2   4   5
        """
        if self.__size == self.__low_bd:  # check if covering is optimal
            repr = ('C(%d, %d, %d) = %d\n' %
                    (self.__v, self.__k, self.__t, self.__size))
        else:
            repr = ('%d <= C(%d, %d, %d) <= %d\n' % (self.__low_bd,
                    self.__v, self.__k, self.__t, self.__size))
        if self.__creator:
            repr += 'Created by: %s\n' % (self.__creator)
        if self.__method:
            repr += 'Method: %s\n' % (self.__method)
        if self.__timestamp:
            repr += 'Submitted on: %s\n' % (self.__timestamp)
        return repr + '\n'.join('  '.join(str(k) for k in block) for block in
                                self.__incidence_structure.blocks())

    def is_covering(self):
        """
        Check all `t`-sets are in fact covered by the blocks of ``self``.

        .. NOTE::

            This is very slow and wasteful of memory.

        EXAMPLES::

            sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
            ....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
            ....:     [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
            sage: C.is_covering()
            True
            sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
            ....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6],
            ....:     [2, 4, 6]], 0, 'not a covering')   # last block altered
            sage: C.is_covering()
            False
        """
        v = self.__v
        k = self.__k
        t = self.__t
        Svt = Combinations(range(v), t)
        Skt = Combinations(range(k), t)
        tset = {}  # tables of t-sets: False = uncovered, True = covered
        for i in Svt:
            tset[tuple(i)] = False
        # mark all t-sets covered by each block
        for a in self.__incidence_structure.blocks():
            for z in Skt:
                y = (a[x] for x in z)
                tset[tuple(y)] = True
        for i in Svt:
            if not tset[tuple(i)]:  # uncovered
                return False
        return True                 # everything was covered

    def v(self):
        """
        Return `v`, the number of points in the covering design.

        EXAMPLES::

            sage: from sage.combinat.designs.covering_design import CoveringDesign
            sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
            ....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
            ....:     [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
            sage: C.v()
            7
        """
        return self.__v

    def k(self):
        """
        Return `k`, the size of blocks of the covering design

        EXAMPLES::

            sage: from sage.combinat.designs.covering_design import CoveringDesign
            sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
            ....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
            ....:     [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
            sage: C.k()
            3
        """
        return self.__k

    def t(self):
        """
        Return `t`, the size of sets which must be covered by the
        blocks of the covering design

        EXAMPLES::

            sage: from sage.combinat.designs.covering_design import CoveringDesign
            sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
            ....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
            ....:     [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
            sage: C.t()
            2
        """
        return self.__t

    def size(self):
        """
        Return the number of blocks in the covering design

        EXAMPLES::

            sage: from sage.combinat.designs.covering_design import CoveringDesign
            sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
            ....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
            ....:     [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
            sage: C.size()
            7
        """
        return self.__size

    def low_bd(self):
        """
        Return a lower bound for the number of blocks a covering
        design with these parameters could have.

        Typically this is the Schonheim bound, but for some parameters
        better bounds have been shown.

        EXAMPLES::

            sage: from sage.combinat.designs.covering_design import CoveringDesign
            sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
            ....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
            ....:     [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
            sage: C.low_bd()
            7
        """
        return self.__low_bd

    def method(self):
        """
        Return the method used to create the covering design.

        This field is optional, and is used in a database to give information
        about how coverings were constructed.

        EXAMPLES::

            sage: from sage.combinat.designs.covering_design import CoveringDesign
            sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
            ....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
            ....:     [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
            sage: C.method()
            'Projective Plane'
        """
        return self.__method

    def creator(self):
        """
        Return the creator of the covering design

        This field is optional, and is used in a database to give
        attribution for the covering design It can refer to the person
        who submitted it, or who originally gave a construction

        EXAMPLES::

            sage: from sage.combinat.designs.covering_design import CoveringDesign
            sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
            ....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6],
            ....:     [2, 4, 5]],0, 'Projective Plane', 'Gino Fano')
            sage: C.creator()
            'Gino Fano'
        """
        return self.__creator

    def timestamp(self):
        """
        Return the time that the covering was submitted to the database

        EXAMPLES::

            sage: from sage.combinat.designs.covering_design import CoveringDesign
            sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
            ....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
            ....:     [2, 3, 6], [2, 4, 5]],0, 'Projective Plane',
            ....:     'Gino Fano', '1892-01-01 00:00:00')
            sage: C.timestamp()  # No exact date known; in Fano's 1892 article
            '1892-01-01 00:00:00'
        """
        return self.__timestamp

    def incidence_structure(self):
        """
        Return the incidence structure of this design, without extra parameters.

        EXAMPLES::

            sage: from sage.combinat.designs.covering_design import CoveringDesign
            sage: C = CoveringDesign(7, 3, 2, 7, range(7), [[0, 1, 2],
            ....:     [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6],
            ....:     [2, 3, 6], [2, 4, 5]], 0, 'Projective Plane')
            sage: D = C.incidence_structure()
            sage: D.ground_set()
            [0, 1, 2, 3, 4, 5, 6]
            sage: D.blocks()
            [[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5],
            [1, 4, 6], [2, 3, 6], [2, 4, 5]]

        """
        return self.__incidence_structure


def best_known_covering_design_www(v, k, t, verbose=False):
    r"""
    Return the best known `(v, k, t)` covering design from an online database.

    This uses the La Jolla Covering Repository, a database
    available at `<https://ljcr.dmgordon.org/cover.html>`_

    INPUT:

    - ``v`` -- integer, the size of the point set for the design

    - ``k`` -- integer, the number of points per block

    - ``t`` -- integer, the size of sets covered by the blocks

    - ``verbose`` -- bool (default: ``False``), print verbose message

    OUTPUT:

    A :class:`CoveringDesign` object representing the ``(v, k, t)``-covering
    design with smallest number of blocks available in the database.

    EXAMPLES::

        sage: from sage.combinat.designs.covering_design import (  # optional - internet
        ....:     best_known_covering_design_www)
        sage: C = best_known_covering_design_www(7, 3, 2)  # optional - internet
        sage: print(C)                                     # optional - internet
        C(7, 3, 2) = 7
        Method: lex covering
        Submitted on: 1996-12-01 00:00:00
        0  1  2
        0  3  4
        0  5  6
        1  3  5
        1  4  6
        2  3  6
        2  4  5

    A ValueError is raised if the ``(v, k, t)`` parameters are not
    found in the database.
    """
    v = int(v)
    k = int(k)
    t = int(t)
    param = "?v=%s&k=%s&t=%s" % (v, k, t)
    url = "https://ljcr.dmgordon.org/cover/get_cover.php" + param
    if verbose:
        print("Looking up the bounds at %s" % url)

    f = urlopen(url)
    try:
        s = bytes_to_str(f.read())
    finally:
        f.close()

    if 'covering not in database' in s:  # not found
        str = "no (%d, %d, %d) covering design in database\n" % (v, k, t)
        raise ValueError(str)
    return sage_eval(s)
