Coverage for local/lib/python2.7/site-packages/sage/data_structures/mutable

Hot-keys on this page

r m x p toggle line displays

j k next/prev highlighted chunk

0 (zero) top of page

1 (one) first highlighted chunk

r"""

Mutable Poset

This module provides a class representing a finite partially ordered

set (poset) for the purpose of being used as a data structure. Thus

the posets introduced in this module are mutable, i.e., elements can

be added and removed from a poset at any time.

To get in touch with Sage's "usual" posets, start with the page

:mod:`Posets <sage.combinat.posets.__init__>` in the reference manual.

.. _mutable_poset_examples:

Examples

========

First Steps

-----------

We start by creating an empty poset. This is simply done by

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: P

poset()

A poset should contain elements, thus let us add them with

sage: P.add(42)

sage: P.add(7)

sage: P.add(13)

sage: P.add(3)

Let us look at the poset again::

sage: P

poset(3, 7, 13, 42)

We see that they elements are sorted using `\leq` which exists on the

integers `\ZZ`. Since this is even a total order, we could have used a

more efficient data structure. Alternatively, we can write

sage: MP([42, 7, 13, 3])

poset(3, 7, 13, 42)

to add several elements at once on construction.

A less boring Example

---------------------

Let us continue with a less boring example. We define the class

sage: class T(tuple):

....: def __le__(left, right):

....: return all(l <= r for l, r in zip(left, right))

It is equipped with a `\leq`-operation such that `a \leq b` if all

entries of `a` are at most the corresponding entry of `b`. For

example, we have

sage: a = T((1,1))

sage: b = T((2,1))

sage: c = T((1,2))

sage: a <= b, a <= c, b <= c

(True, True, False)

The last comparison gives ``False``, since the comparison of the

first component checks whether `2 \leq 1`.

Now, let us add such elements to a poset::

sage: Q = MP([T((1, 1)), T((3, 3)), T((4, 1)),

....: T((3, 2)), T((2, 3)), T((2, 2))]); Q

poset((1, 1), (2, 2), (2, 3), (3, 2), (3, 3), (4, 1))

In the representation above, the elements are sorted topologically,

smallest first. This does not (directly) show more structural

information. We can overcome this and display a "wiring layout" by

typing::

sage: print(Q.repr_full(reverse=True))

poset((3, 3), (2, 3), (3, 2), (2, 2), (4, 1), (1, 1))

+-- oo

| +-- no successors

| +-- predecessors: (3, 3), (4, 1)

+-- (3, 3)

| +-- successors: oo

| +-- predecessors: (2, 3), (3, 2)

+-- (2, 3)

| +-- successors: (3, 3)

| +-- predecessors: (2, 2)

+-- (3, 2)

| +-- successors: (3, 3)

| +-- predecessors: (2, 2)

+-- (2, 2)

| +-- successors: (2, 3), (3, 2)

| +-- predecessors: (1, 1)

+-- (4, 1)

| +-- successors: oo

| +-- predecessors: (1, 1)

+-- (1, 1)

| +-- successors: (2, 2), (4, 1)

| +-- predecessors: null

+-- null

| +-- successors: (1, 1)

| +-- no predecessors

Note that we use ``reverse=True`` to let the elements appear from

largest (on the top) to smallest (on the bottom).

If you look at the output above, you'll see two additional elements,

namely ``oo`` (`\infty`) and ``null`` (`\emptyset`). So what are these

strange animals? The answer is simple and maybe you can guess it

already. The `\infty`-element is larger than every other element,

therefore a successor of the maximal elements in the poset. Similarly,

the `\emptyset`-element is smaller than any other element, therefore a

predecessor of the poset's minimal elements. Both do not have to scare

us; they are just there and sometimes useful.

AUTHORS:

- Daniel Krenn (2015)

ACKNOWLEDGEMENT:

- Daniel Krenn is supported by the Austrian Science Fund (FWF): P 24644-N26.

Classes and their Methods

=========================

"""

#*****************************************************************************

# 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 print_function

from six import itervalues

from sage.structure.sage_object import SageObject

class MutablePosetShell(SageObject):

r"""

A shell for an element of a :class:`mutable poset <MutablePoset>`.

INPUT:

- ``poset`` -- the poset to which this shell belongs.

- ``element`` -- the element which should be

contained/encapsulated in this shell.

OUTPUT:

A shell for the given element.

.. NOTE::

If the :meth:`element` of a shell is ``None``, then this

element is considered as "special" (see :meth:`is_special`).

There are two special elements, namely

- a ``'null'`` (an element smaller than each other element;

it has no predecessors) and

- an ``'oo'`` (an element larger than each other element;

it has no successors).

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: P.add(66)

sage: P

poset(66)

sage: s = P.shell(66)

sage: type(s)

.. SEEALSO::

:class:`MutablePoset`

"""

def __init__(self, poset, element):

r"""

See :class:`MutablePosetShell` for details.

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: from sage.data_structures.mutable_poset import MutablePosetShell

sage: MutablePosetShell(P, (1, 2))

(1, 2)

"""

self._poset_ = poset

self._element_ = element

self._key_ = self.poset.get_key(element)

self._predecessors_ = set()

self._successors_ = set()

super(MutablePosetShell, self).__init__()

@property

def poset(self):

r"""

The poset to which this shell belongs.

.. SEEALSO::

:class:`MutablePoset`

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: from sage.data_structures.mutable_poset import MutablePosetShell

sage: e = MutablePosetShell(P, (1, 2))

sage: e.poset is P

True

"""

return self._poset_

@property

def element(self):

r"""

The element contained in this shell.

.. SEEALSO::

:meth:`key`,

:class:`MutablePoset`.

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: from sage.data_structures.mutable_poset import MutablePosetShell

sage: e = MutablePosetShell(P, (1, 2))

sage: e.element

(1, 2)

"""

return self._element_

@property

def key(self):

r"""

The key of the element contained in this shell.

The key of an element is determined by the mutable poset (the

parent) via the ``key``-function (see construction of a

:class:`MutablePoset`).

.. SEEALSO::

:meth:`element`,

:class:`MutablePoset`.

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: from sage.data_structures.mutable_poset import MutablePosetShell

sage: P = MP()

sage: e = MutablePosetShell(P, (1, 2))

sage: e.key

(1, 2)

sage: Q = MP(key=lambda k: k[0])

sage: f = MutablePosetShell(Q, (1, 2))

sage: f.key

Test the caching of the key::

sage: def k(k):

....: print('key %s' % (k,))

....: return k

sage: R = MP(key=k)

sage: h = MutablePosetShell(R, (1, 2))

key (1, 2)

sage: h.key; h.key

(1, 2)

"""

return self._key_

def predecessors(self, reverse=False):

r"""

Return the predecessors of this shell.

INPUT:

- ``reverse`` -- (default: ``False``) if set, then return

successors instead.

OUTPUT:

A set.

.. SEEALSO::

:meth:`successors`,

:class:`MutablePoset`.

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: from sage.data_structures.mutable_poset import MutablePosetShell

sage: e = MutablePosetShell(P, (1, 2))

sage: e.predecessors()

set()

"""

if reverse:

return self._successors_

return self._predecessors_

def successors(self, reverse=False):

r"""

Return the successors of this shell.

INPUT:

- ``reverse`` -- (default: ``False``) if set, then return

predecessors instead.

OUTPUT:

A set.

.. SEEALSO::

:meth:`predecessors`,

:class:`MutablePoset`.

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: from sage.data_structures.mutable_poset import MutablePosetShell

sage: e = MutablePosetShell(P, (1, 2))

sage: e.successors()

set()

"""

if reverse:

return self._predecessors_

return self._successors_

def is_special(self):

r"""

Return whether this shell contains either the null-element, i.e., the

element smaller than any possible other element or the

infinity-element, i.e., the element larger than any possible

other element.

INPUT:

Nothing.

OUTPUT:

``True`` or ``False``.

.. SEEALSO::

:meth:`is_null`,

:meth:`is_oo`,

:class:`MutablePoset`.

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: P.null.is_special()

True

sage: P.oo.is_special()

True

"""

return self.element is None

def is_null(self):

r"""

Return whether this shell contains the null-element, i.e., the element

smaller than any possible other element.

OUTPUT:

``True`` or ``False``.

.. SEEALSO::

:meth:`is_special`,

:meth:`is_oo`,

:meth:`MutablePoset.null`,

:class:`MutablePoset`.

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: P.null.is_null()

True

sage: P.oo.is_null()

False

"""

return self.element is None and not self.predecessors()

def is_oo(self):

r"""

Return whether this shell contains the infinity-element, i.e., the element

larger than any possible other element.

OUTPUT:

``True`` or ``False``.

.. SEEALSO::

:meth:`is_null`,

:meth:`is_special`,

:meth:`MutablePoset.oo`,

:class:`MutablePoset`.

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: P.null.is_oo()

False

sage: P.oo.is_oo()

True

"""

return self.element is None and not self.successors()

def _repr_(self):

r"""

Return the representation of this shell.

INPUT:

Nothing.

OUTPUT:

A string.

.. NOTE::

If the :meth:`element` of this shell is not ``None``,

this method returns the respective representation string.

Otherwise, ``'null'`` or ``'oo'`` are returned,

depending on the non-existence of predecessors or

successors, respectively.

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: from sage.data_structures.mutable_poset import MutablePosetShell

sage: repr(MutablePosetShell(P, (1, 2))) # indirect doctest

'(1, 2)'

sage: repr(P.null) # indirect doctest

'null'

sage: repr(P.oo) # indirect doctest

'oo'

"""

if self.is_null():

return 'null'

elif self.is_oo():

return 'oo'

else:

return repr(self.element)

def __hash__(self):

r"""

Return the hash of this shell.

INPUT:

Nothing.

OUTPUT:

A hash value.

This returns the hash value of the key of the element

contained in this shell.

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: from sage.data_structures.mutable_poset import MutablePosetShell

sage: hash(MutablePosetShell(P, (1, 2))) == hash((1, 2))

True

"""

return hash(self.key)

def le(self, other, reverse=False):

r"""

Return whether this shell is less than or equal to ``other``.

INPUT:

- ``other`` -- a shell.

- ``reverse`` -- (default: ``False``) if set, then return

whether this shell is greater than or equal to ``other``.

OUTPUT:

``True`` or ``False``.

.. NOTE::

The comparison of the shells is based on the comparison

of the keys of the elements contained in the shells,

except for special shells (see :class:`MutablePosetShell`).

.. SEEALSO::

:meth:`eq`,

:class:`MutablePoset`.

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: from sage.data_structures.mutable_poset import MutablePosetShell

sage: e = MutablePosetShell(P, (1, 2))

sage: z = P.null

sage: oo = P.oo

sage: z <= e # indirect doctest

True

sage: e <= oo # indirect doctest

True

sage: z <= oo # indirect doctest

True

sage: oo <= z # indirect doctest

False

sage: oo <= e # indirect doctest

False

sage: e <= z # indirect doctest

False

sage: z <= z # indirect doctest

True

sage: oo <= oo # indirect doctest

True

sage: e <= e # indirect doctest

True

sage: z.le(e, reverse=True)

False

sage: e.le(oo, reverse=True)

False

sage: z.le(oo, reverse=True)

False

sage: oo.le(z, reverse=True)

True

sage: oo.le(e, reverse=True)

True

sage: e.le(z, reverse=True)

True

sage: z.le(z, reverse=True)

True

sage: oo.le(oo, reverse=True)

True

sage: e.le(e, reverse=True)

True

"""

if reverse:

return other.le(self, reverse=False)

if self.element is None:

if not self._predecessors_:

# null on the left

return True

else:

# oo on the left

if other.element is None:

# null or oo on the right

return not other._successors_

else:

# not null, not oo on the right

return False

elif other.element is None:

# null/oo on the right

return not other._successors_

return self.key <= other.key

__le__ = le

def eq(self, other):

r"""

Return whether this shell is equal to ``other``.

INPUT:

- ``other`` -- a shell.

OUTPUT:

``True`` or ``False``.

.. NOTE::

This method compares the keys of the elements contained

in the (non-special) shells. In particular,

elements/shells with the same key are considered as equal.

.. SEEALSO::

:meth:`le`,

:class:`MutablePoset`.

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: from sage.data_structures.mutable_poset import MutablePosetShell

sage: e = MutablePosetShell(P, (1, 2))

sage: f = MutablePosetShell(P, (2, 1))

sage: z = P.null

sage: oo = P.oo

sage: z == z

True

sage: oo == oo

True

sage: e == e

True

sage: e == f

False

sage: z == e

False

sage: e == oo

False

sage: oo == z

False

Comparing elements in different mutable posets is possible; their

shells are equal if their elements are::

sage: S = MP([42]); s = S.shell(42)

sage: T = MP([42]); t = T.shell(42)

sage: s == t

True

sage: S.oo == T.oo

True

"""

if self.element is None and other.element is None:

return self.is_null() == other.is_null()

return self.key == other.key

__eq__ = eq

def _copy_all_linked_(self, memo, poset, mapping):

r"""

Return a copy of this shell. All shells linked to this shell

are copied as well.

This is a helper function for :meth:`MutablePoset.copy`.

INPUT:

- ``memo`` -- a dictionary which assigns to the id of the

calling shell to a copy of it.

- ``poset`` -- the poset to which the newly created shells

belongs. Note that the elements are not inserted into

``poset``; this is done in the calling method

:meth:`MutablePoset._copy_shells_`.

- ``mapping`` -- a function which is applied on each of the elements.

OUTPUT:

A new shell.

.. SEEALSO::

:meth:`MutablePoset.copy`,

:class:`MutablePoset`.

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: Q = MP()

sage: memo = {}

sage: z = P.null._copy_all_linked_(memo, Q, lambda e: e)

sage: z.poset is Q

True

sage: oo = z.successors().pop()

sage: oo.is_oo()

True

Note that :meth:`_copy_all_linked_` does not change the mutable

poset ``Q`` (this is done in the calling method

:meth:`MutablePoset._copy_shells_`). Thus we have

sage: oo is Q.oo

False

"""

try:

return memo[id(self)]

except KeyError:

pass

new = self.__class__(poset, mapping(self.element)

if self.element is not None else None)

memo[id(self)] = new

for reverse in (False, True):

for e in self.successors(reverse):

new.successors(reverse).add(e._copy_all_linked_(memo, poset, mapping))

return new

def lower_covers(self, shell, reverse=False):

r"""

Return the lower covers of the specified ``shell``;

the search is started at this (``self``) shell.

A lower cover of `x` is an element `y` of the poset

such that `y < x` and there is no element `z` of the poset

so that `y < z < x`.

INPUT:

- ``shell`` -- the shell for which to find the covering shells.

There is no restriction of ``shell`` being contained in the poset.

If ``shell`` is contained in the poset, then use the more efficient

methods :meth:`predecessors` and :meth:`successors`.

- ``reverse`` -- (default: ``False``) if set, then find

the upper covers (see also :meth:`upper_covers`)

instead of the lower covers.

OUTPUT:

A set of :class:`shells <MutablePosetShell>`.

.. NOTE::

Suppose ``reverse`` is ``False``. This method starts at

the calling shell (``self``) and searches towards ``'oo'``.

Thus, only shells which are (not necessarily

direct) successors of this shell are considered.

If ``reverse`` is ``True``, then the reverse direction is

taken.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: class T(tuple):

....: def __le__(left, right):

....: return all(l <= r for l, r in zip(left, right))

sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)),

....: T((4, 4)), T((1, 2)), T((2, 2))])

sage: e = P.shell(T((2, 2))); e

(2, 2)

sage: sorted(P.null.lower_covers(e),

....: key=lambda c: repr(c.element))

[(1, 2), (2, 1)]

sage: set(_) == e.predecessors()

True

sage: sorted(P.oo.upper_covers(e),

....: key=lambda c: repr(c.element))

[(4, 4)]

sage: set(_) == e.successors()

True

sage: Q = MP([T((3, 2))])

sage: f = next(Q.shells())

sage: sorted(P.null.lower_covers(f),

....: key=lambda c: repr(c.element))

[(2, 2)]

sage: sorted(P.oo.upper_covers(f),

....: key=lambda c: repr(c.element))

[(4, 4)]

.. SEEALSO::

:meth:`upper_covers`,

:meth:`predecessors`,

:meth:`successors`,

:class:`MutablePoset`.

"""

if self == shell:

return set()

covers = set().union(*(e.lower_covers(shell, reverse)

for e in self.successors(reverse)

if e.le(shell, reverse)))

return covers or set([self])

def upper_covers(self, shell, reverse=False):

r"""

Return the upper covers of the specified ``shell``;

the search is started at this (``self``) shell.

An upper cover of `x` is an element `y` of the poset

such that `x < y` and there is no element `z` of the poset

so that `x < z < y`.

INPUT:

- ``shell`` -- the shell for which to find the covering shells.

There is no restriction of ``shell`` being contained in the poset.

If ``shell`` is contained in the poset, then use the more efficient

methods :meth:`predecessors` and :meth:`successors`.

- ``reverse`` -- (default: ``False``) if set, then find

the lower covers (see also :meth:`lower_covers`)

instead of the upper covers.

OUTPUT:

A set of :class:`shells <MutablePosetShell>`.

.. NOTE::

Suppose ``reverse`` is ``False``. This method starts at

the calling shell (``self``) and searches towards ``'null'``.

Thus, only shells which are (not necessarily

direct) predecessors of this shell are considered.

If ``reverse`` is ``True``, then the reverse direction is

taken.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: class T(tuple):

....: def __le__(left, right):

....: return all(l <= r for l, r in zip(left, right))

sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)),

....: T((4, 4)), T((1, 2)), T((2, 2))])

sage: e = P.shell(T((2, 2))); e

(2, 2)

sage: sorted(P.null.lower_covers(e),

....: key=lambda c: repr(c.element))

[(1, 2), (2, 1)]

sage: set(_) == e.predecessors()

True

sage: sorted(P.oo.upper_covers(e),

....: key=lambda c: repr(c.element))

[(4, 4)]

sage: set(_) == e.successors()

True

sage: Q = MP([T((3, 2))])

sage: f = next(Q.shells())

sage: sorted(P.null.lower_covers(f),

....: key=lambda c: repr(c.element))

[(2, 2)]

sage: sorted(P.oo.upper_covers(f),

....: key=lambda c: repr(c.element))

[(4, 4)]

.. SEEALSO::

:meth:`predecessors`,

:meth:`successors`,

:class:`MutablePoset`.

"""

return self.lower_covers(shell, not reverse)

def _iter_depth_first_visit_(self, marked,

reverse=False, key=None,

condition=None):

r"""

Return an iterator over all shells in depth first order.

This is a helper function for :meth:`iter_depth_first`.

INPUT:

- ``marked`` -- a set in which marked shells are stored.

- ``reverse`` -- (default: ``False``) if set, reverses the

order, i.e., ``False`` searches towards ``'oo'`` and

``True`` searches towards ``'null'``.

- ``key`` -- (default: ``None``) a function used for sorting

the direct successors of a shell (used in case of a

tie). If this is ``None``, no sorting occurs.

- ``condition`` -- (default: ``None``) a function mapping a

shell to ``True`` (include in iteration) or ``False`` (do

not include). ``None`` is equivalent to a function returning

always ``True``. Note that the iteration does not go beyond a

not included shell.

OUTPUT:

An iterator.

.. NOTE::

The depth first search starts at this (``self``) shell. Thus

only this shell and shells greater than (in case of

``reverse=False``) this shell are visited.

.. SEEALSO::

:meth:`iter_depth_first`,

:meth:`iter_topological`,

:class:`MutablePoset`.

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: P.add(42)

sage: P.add(5)

sage: marked = set()

sage: list(P.oo._iter_depth_first_visit_(marked, reverse=True))

[oo, 42, 5, null]

"""

if (condition is not None and

not self.is_special() and not condition(self)):

return

if self in marked:

return

marked.add(self)

yield self

S = self.successors(reverse)

if key is not None:

S = sorted(S, key=key)

for shell in S:

for e in shell._iter_depth_first_visit_(marked, reverse,

key, condition):

yield e

def iter_depth_first(self, reverse=False, key=None, condition=None):

r"""

Iterate over all shells in depth first order.

INPUT:

- ``reverse`` -- (default: ``False``) if set, reverses the

order, i.e., ``False`` searches towards ``'oo'`` and

``True`` searches towards ``'null'``.

- ``key`` -- (default: ``None``) a function used for sorting

the direct successors of a shell (used in case of a

tie). If this is ``None``, no sorting occurs.

- ``condition`` -- (default: ``None``) a function mapping a

shell to ``True`` (include in iteration) or ``False`` (do

not include). ``None`` is equivalent to a function returning

always ``True``. Note that the iteration does not go beyond a

not included shell.

OUTPUT:

An iterator.

.. NOTE::

The depth first search starts at this (``self``) shell. Thus

only this shell and shells greater than (in case of

``reverse=False``) this shell are visited.

ALGORITHM:

See :wikipedia:`Depth-first_search`.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: class T(tuple):

....: def __le__(left, right):

....: return all(l <= r for l, r in zip(left, right))

sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)),

....: T((4, 4)), T((1, 2)), T((2, 2))])

sage: list(P.null.iter_depth_first(reverse=False, key=repr))

[null, (1, 1), (1, 2), (1, 3), (4, 4), oo, (2, 2), (2, 1)]

sage: list(P.oo.iter_depth_first(reverse=True, key=repr))

[oo, (4, 4), (1, 3), (1, 2), (1, 1), null, (2, 2), (2, 1)]

sage: list(P.null.iter_depth_first(

....: condition=lambda s: s.element[0] == 1))

[null, (1, 1), (1, 2), (1, 3)]

.. SEEALSO::

:meth:`iter_topological`,

:class:`MutablePoset`.

"""

marked = set()

return self._iter_depth_first_visit_(marked, reverse, key, condition)

def _iter_topological_visit_(self, marked,

reverse=False, key=None,

condition=None):

r"""

Return an iterator over all shells in topological order.

This is a helper function for :meth:`iter_topological`.

INPUT:

- ``marked`` -- a set in which marked shells are stored.

- ``reverse`` -- (default: ``False``) if set, reverses the

order, i.e., ``False`` searches towards ``'oo'`` and

``True`` searches towards ``'null'``.

- ``key`` -- (default: ``None``) a function used for sorting

the direct predecessors of a shell (used in case of a

tie). If this is ``None``, no sorting occurs.

- ``condition`` -- (default: ``None``) a function mapping a

shell to ``True`` (include in iteration) or ``False`` (do

not include). ``None`` is equivalent to a function returning

always ``True``. Note that the iteration does not go beyond a

not included shell.

OUTPUT:

An iterator.

.. NOTE::

The topological search will only find shells smaller than

(in case of ``reverse=False``)

or equal to this (``self``) shell. This is in contrast to

:meth:`iter_depth_first`.

.. SEEALSO::

:meth:`iter_depth_first`,

:meth:`iter_topological`,

:class:`MutablePoset`.

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: P.add(42)

sage: P.add(5)

sage: marked = set()

sage: list(P.null._iter_topological_visit_(marked, reverse=True))

[oo, 42, 5, null]

"""

if (condition is not None and

not self.is_special() and not condition(self)):

return

if self in marked:

return

marked.add(self)

S = self.predecessors(reverse)

if key is not None and len(S) > 1:

S = sorted(S, key=key)

for shell in S:

for e in shell._iter_topological_visit_(marked, reverse,

key, condition):

yield e

yield self

def iter_topological(self, reverse=False, key=None, condition=None):

r"""

Iterate over all shells in topological order.

INPUT:

- ``reverse`` -- (default: ``False``) if set, reverses the

order, i.e., ``False`` searches towards ``'oo'`` and

``True`` searches towards ``'null'``.

- ``key`` -- (default: ``None``) a function used for sorting

the direct predecessors of a shell (used in case of a

tie). If this is ``None``, no sorting occurs.

- ``condition`` -- (default: ``None``) a function mapping a

shell to ``True`` (include in iteration) or ``False`` (do

not include). ``None`` is equivalent to a function returning

always ``True``. Note that the iteration does not go beyond a

not included shell.

OUTPUT:

An iterator.

.. NOTE::

The topological search will only find shells smaller than

(in case of ``reverse=False``)

or equal to this (``self``) shell. This is in contrast to

:meth:`iter_depth_first`.

ALGORITHM:

Here a simplified version of the algorithm found in [Tar1976]_

and [CLRS2001]_ is used. See also

:wikipedia:`Topological_sorting`.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: class T(tuple):

....: def __le__(left, right):

....: return all(l <= r for l, r in zip(left, right))

sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)),

....: T((4, 4)), T((1, 2)), T((2, 2))])

sage: for e in P.shells_topological(include_special=True,

....: reverse=True):

....: print(e)

....: print(list(e.iter_topological(reverse=True, key=repr)))

[oo]

(4, 4)

[oo, (4, 4)]

(1, 3)

[oo, (4, 4), (1, 3)]

(2, 2)

[oo, (4, 4), (2, 2)]

(1, 2)

[oo, (4, 4), (1, 3), (2, 2), (1, 2)]

(2, 1)

[oo, (4, 4), (2, 2), (2, 1)]

(1, 1)

[oo, (4, 4), (1, 3), (2, 2), (1, 2), (2, 1), (1, 1)]

null

[oo, (4, 4), (1, 3), (2, 2), (1, 2), (2, 1), (1, 1), null]

sage: for e in P.shells_topological(include_special=True,

....: reverse=True):

....: print(e)

....: print(list(e.iter_topological(reverse=False, key=repr)))

[null, (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (4, 4), oo]

(4, 4)

[null, (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (4, 4)]

(1, 3)

[null, (1, 1), (1, 2), (1, 3)]

(2, 2)

[null, (1, 1), (1, 2), (2, 1), (2, 2)]

(1, 2)

[null, (1, 1), (1, 2)]

(2, 1)

[null, (1, 1), (2, 1)]

(1, 1)

[null, (1, 1)]

null

[null]

sage: list(P.null.iter_topological(

....: reverse=True, condition=lambda s: s.element[0] == 1))

[(1, 3), (1, 2), (1, 1), null]

.. SEEALSO::

:meth:`iter_depth_first`,

:meth:`MutablePoset.shells_topological`,

:meth:`MutablePoset.elements_topological`,

:meth:`MutablePoset.keys_topological`,

:class:`MutablePoset`.

"""

marked = set()

return self._iter_topological_visit_(marked, reverse, key, condition)

def merge(self, element, check=True, delete=True):

r"""

Merge the given element with the element contained in this

shell.

INPUT:

- ``element`` -- an element (of the poset).

- ``check`` -- (default: ``True``) if set, then the

``can_merge``-function of :class:`MutablePoset` determines

whether the merge is possible. ``can_merge`` is ``None`` means

that this check is always passed.

- ``delete`` -- (default: ``True``) if set, then ``element``

is removed from the poset after the merge.

OUTPUT:

Nothing.

.. NOTE::

This operation depends on the parameters ``merge`` and

``can_merge`` of the :class:`MutablePoset` this shell is

contained in. These parameters are defined when the poset

is constructed.

.. NOTE::

If the ``merge`` function returns ``None``, then this shell

is removed from the poset.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: def add(left, right):

....: return (left[0], ''.join(sorted(left[1] + right[1])))

sage: def can_add(left, right):

....: return left[0] <= right[0]

sage: P = MP([(1, 'a'), (3, 'b'), (2, 'c'), (4, 'd')],

....: key=lambda c: c[0], merge=add, can_merge=can_add)

sage: P

poset((1, 'a'), (2, 'c'), (3, 'b'), (4, 'd'))

sage: P.shell(2).merge((3, 'b'))

sage: P

poset((1, 'a'), (2, 'bc'), (4, 'd'))

.. SEEALSO::

:meth:`MutablePoset.merge`,

:class:`MutablePoset`.

TESTS::

sage: MP([2], merge=operator.add,

....: can_merge=lambda _, __: False).shell(2).merge(1)

Traceback (most recent call last):

...

RuntimeError: Cannot merge 2 with 1.

"""

poset = self.poset

if poset._merge_ is None:

# poset._merge_ is None means no merge (poset._merge_ simply

# returns its first input argument).

return

self_element = self.element

if check:

if not poset._can_merge_(self_element, element):

raise RuntimeError('Cannot merge %s with %s.' %

(self_element, element))

new = poset._merge_(self_element, element)

if new is None:

poset.discard(poset.get_key(self.element))

else:

self._element_ = new

if delete:

poset.remove(poset.get_key(element))

# *****************************************************************************

def is_MutablePoset(P):

r"""

Test whether ``P`` inherits from :class:`MutablePoset`.

.. SEEALSO::

:class:`MutablePoset`

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: from sage.data_structures.mutable_poset import is_MutablePoset

sage: P = MP()

sage: is_MutablePoset(P)

True

"""

return isinstance(P, MutablePoset)

class MutablePoset(SageObject):

r"""

A data structure that models a mutable poset (partially ordered

set).

INPUT:

- ``data`` -- data from which to construct the poset. It can be

any of the following:

#. ``None`` (default), in which case an empty poset is created,

#. a :class:`MutablePoset`, which will be copied during creation,

#. an iterable, whose elements will be in the poset.

- ``key`` -- a function which maps elements to keys. If ``None``

(default), this is the identity, i.e., keys are equal to their

elements.

Two elements with the same keys are considered as equal; so only

one of these two elements can be in the poset.

This ``key`` is not used for sorting (in contrast to

sorting-functions, e.g. ``sorted``).

- ``merge`` -- a function which merges its second argument (an

element) to its first (again an element) and returns the result

(as an element). If the return value is ``None``, the element is

removed from the poset.

This hook is called by :meth:`merge`. Moreover it is used during

:meth:`add` when an element (more precisely its key) is already

in this poset.

``merge`` is ``None`` (default) is equivalent to ``merge``

returning its first argument. Note that it is not allowed that the

key of the returning element differs from the key of the first

input parameter. This means ``merge`` must not change the

position of the element in the poset.

- ``can_merge`` -- a function which checks whether its second argument

can be merged to its first.

This hook is called by :meth:`merge`. Moreover it is used during

:meth:`add` when an element (more precisely its key) is already

in this poset.

``can_merge`` is ``None`` (default) is equivalent to ``can_merge``

returning ``True`` in all cases.

OUTPUT:

A mutable poset.

You can find a short introduction and examples

:mod:`here <sage.data_structures.mutable_poset>`.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

We illustrate the different input formats

#. No input::

sage: A = MP(); A

poset()

#. A :class:`MutablePoset`::

sage: B = MP(A); B

poset()

sage: B.add(42)

sage: C = MP(B); C

poset(42)

#. An iterable::

sage: C = MP([5, 3, 11]); C

poset(3, 5, 11)

.. SEEALSO::

:class:`MutablePosetShell`.

"""

def __init__(self, data=None, key=None, merge=None, can_merge=None):

r"""

See :class:`MutablePoset` for details.

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: MP()

poset()

sage: P = MP()

sage: P.add(42)

sage: MP(P)

poset(42)

sage: MP([3, 5, 7])

poset(3, 5, 7)

sage: MP(33)

Traceback (most recent call last):

...

TypeError: 33 is not iterable; do not know what to do with it.

"""

if is_MutablePoset(data):

if key is not None:

raise TypeError('Cannot use key when data is a poset.')

self._copy_shells_(data, lambda e: e)

else:

self.clear()

if key is None:

self._key_ = lambda k: k

else:

self._key_ = key

self._merge_ = merge

if can_merge is None:

self._can_merge_ = lambda _, __: True

else:

self._can_merge_ = can_merge

if data is not None:

try:

it = iter(data)

except TypeError:

raise TypeError('%s is not iterable; do not know what to '

'do with it.' % (data,))

self.union_update(it)

super(MutablePoset, self).__init__()

def clear(self):

r"""

Remove all elements from this poset.

INPUT:

Nothing.

OUTPUT:

Nothing.

.. SEEALSO::

:meth:`discard`,

:meth:`pop`,

:meth:`remove`.

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: P.add(42); P

poset(42)

sage: P.clear()

sage: print(P.repr_full())

poset()

+-- null

| +-- no predecessors

| +-- successors: oo

+-- oo

| +-- predecessors: null

| +-- no successors

"""

self._null_ = MutablePosetShell(self, None)

self._oo_ = MutablePosetShell(self, None)

self._null_.successors().add(self._oo_)

self._oo_.predecessors().add(self._null_)

self._shells_ = {}

def __len__(self):

r"""

Return the number of elements contained in this poset.

INPUT:

Nothing.

OUTPUT:

An integer.

.. NOTE::

The special elements ``'null'`` and ``'oo'`` are not counted.

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: len(P) # indirect doctest

sage: bool(P)

False

sage: P.add(42)

sage: len(P)

sage: bool(P)

True

"""

return len(self._shells_)

@property

def null(self):

r"""

The shell `\emptyset` whose element is smaller than any

other element.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: z = P.null; z

null

sage: z.is_null()

True

.. SEEALSO::

:meth:`oo`,

:meth:`MutablePosetShell.is_null`,

:meth:`MutablePosetShell.is_special`.

"""

return self._null_

@property

def oo(self):

r"""

The shell `\infty` whose element is larger than any other

element.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: oo = P.oo; oo

sage: oo.is_oo()

True

.. SEEALSO::

:meth:`null`,

:meth:`MutablePosetShell.is_oo`,

:meth:`MutablePosetShell.is_special`.

"""

return self._oo_

def shell(self, key):

r"""

Return the shell of the element corresponding to ``key``.

INPUT:

``key`` -- the key of an object.

OUTPUT:

An instance of :class:`MutablePosetShell`.

.. NOTE::

Each element is contained/encapsulated in a shell inside

the poset.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: P.add(42)

sage: e = P.shell(42); e

sage: type(e)

.. SEEALSO::

:meth:`element`,

:meth:`get_key`.

"""

return self._shells_[key]

def element(self, key):

r"""

Return the element corresponding to ``key``.

INPUT:

``key`` -- the key of an object.

OUTPUT:

An object.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: P.add(42)

sage: e = P.element(42); e

sage: type(e)

.. SEEALSO::

:meth:`shell`,

:meth:`get_key`.

"""

return self.shell(key).element

def get_key(self, element):

r"""

Return the key corresponding to the given element.

INPUT:

- ``element`` -- an object.

OUTPUT:

An object (the key of ``element``).

.. SEEALSO::

:meth:`element`,

:meth:`shell`.

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: P.get_key(None) is None

True

sage: P.get_key((1, 2))

(1, 2)

sage: Q = MP(key=lambda k: k[0])

sage: Q.get_key((1, 2))

"""

if element is None:

return None

return self._key_(element)

def _copy_shells_(self, other, mapping):

r"""

Copy shells from another poset.

INPUT:

- ``other`` -- the mutable poset from which the shells

should be copied to this poset.

- ``mapping`` -- a function that is applied to each element.

OUTPUT:

Nothing.

.. SEEALSO::

:meth:`copy`

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: class T(tuple):

....: def __le__(left, right):

....: return all(l <= r for l, r in zip(left, right))

sage: P = MP()

sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)),

....: T((4, 4)), T((1, 2))])

sage: Q = MP()

sage: Q._copy_shells_(P, lambda e: e)

sage: P.repr_full() == Q.repr_full()

True

"""

from copy import copy

self._key_ = copy(other._key_)

self._merge_ = copy(other._merge_)

self._can_merge_ = copy(other._can_merge_)

memo = {}

self._null_ = other._null_._copy_all_linked_(memo, self, mapping)

self._oo_ = memo[id(other._oo_)]

self._shells_ = {f.key: f for f in iter(memo[id(e)] for e in

itervalues(other._shells_))}

def copy(self, mapping=None):

r"""

Create a shallow copy.

INPUT:

- ``mapping`` -- a function which is applied on each of the elements.

OUTPUT:

A poset with the same content as ``self``.

.. SEEALSO::

:meth:`map`,

:meth:`mapped`.

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: class T(tuple):

....: def __le__(left, right):

....: return all(l <= r for l, r in zip(left, right))

sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)),

....: T((4, 4)), T((1, 2))])

sage: Q = copy(P) # indirect doctest

sage: P.repr_full() == Q.repr_full()

True

"""

if mapping is None:

def mapping(element):

return element

new = self.__class__()

new._copy_shells_(self, mapping)

return new

__copy__ = copy

def shells(self, include_special=False):

r"""

Return an iterator over all shells.

INPUT:

- ``include_special`` -- (default: ``False``) if set, then

including shells containing a smallest element (`\emptyset`)

and a largest element (`\infty`).

OUTPUT:

An iterator.

.. NOTE::

Each element is contained/encapsulated in a shell inside

the poset.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: tuple(P.shells())

()

sage: tuple(P.shells(include_special=True))

(null, oo)

.. SEEALSO::

:meth:`shells_topological`,

:meth:`elements`,

:meth:`elements_topological`,

:meth:`keys`,

:meth:`keys_topological`,

:meth:`MutablePosetShell.iter_depth_first`,

:meth:`MutablePosetShell.iter_topological`.

"""

if include_special:

yield self.null

for e in itervalues(self._shells_):

yield e

if include_special:

yield self.oo

def shells_topological(self, include_special=False,

reverse=False, key=None):

r"""

Return an iterator over all shells in topological order.

INPUT:

- ``include_special`` -- (default: ``False``) if set, then

including shells containing a smallest element (`\emptyset`)

and a largest element (`\infty`).

- ``reverse`` -- (default: ``False``) -- if set, reverses the

order, i.e., ``False`` gives smallest elements first,

``True`` gives largest first.

- ``key`` -- (default: ``None``) a function used for sorting

the direct successors of a shell (used in case of a tie). If

this is ``None``, then the successors are sorted according

to their representation strings.

OUTPUT:

An iterator.

.. NOTE::

Each element is contained/encapsulated in a shell inside

the poset.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: class T(tuple):

....: def __le__(left, right):

....: return all(l <= r for l, r in zip(left, right))

sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)),

....: T((4, 4)), T((1, 2)), T((2, 2))])

sage: list(P.shells_topological())

[(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (4, 4)]

sage: list(P.shells_topological(reverse=True))

[(4, 4), (1, 3), (2, 2), (1, 2), (2, 1), (1, 1)]

sage: list(P.shells_topological(include_special=True))

[null, (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (4, 4), oo]

sage: list(P.shells_topological(

....: include_special=True, reverse=True))

[oo, (4, 4), (1, 3), (2, 2), (1, 2), (2, 1), (1, 1), null]

.. SEEALSO::

:meth:`shells`,

:meth:`elements`,

:meth:`elements_topological`,

:meth:`keys`,

:meth:`keys_topological`,

:meth:`MutablePosetShell.iter_depth_first`,

:meth:`MutablePosetShell.iter_topological`.

"""

if key is None:

key = repr

shell = self.oo if not reverse else self.null

return iter(e for e in shell.iter_topological(reverse, key)

if include_special or not e.is_special())

def elements(self, **kwargs):

r"""

Return an iterator over all elements.

INPUT:

- ``kwargs`` -- arguments are passed to :meth:`shells`.

OUTPUT:

An iterator.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP([3, 42, 7])

sage: [(v, type(v)) for v in sorted(P.elements())]

[(3, <type 'sage.rings.integer.Integer'>),

(7, <type 'sage.rings.integer.Integer'>),

(42, <type 'sage.rings.integer.Integer'>)]

Note that

sage: it = iter(P)

sage: sorted(it)

[3, 7, 42]

returns all elements as well.

.. SEEALSO::

:meth:`shells`,

:meth:`shells_topological`,

:meth:`elements_topological`,

:meth:`keys`,

:meth:`keys_topological`,

:meth:`MutablePosetShell.iter_depth_first`,

:meth:`MutablePosetShell.iter_topological`.

"""

for shell in self.shells(**kwargs):

yield shell.element

__iter__ = elements

def elements_topological(self, **kwargs):

r"""

Return an iterator over all elements in topological order.

INPUT:

- ``kwargs`` -- arguments are passed to :meth:`shells_topological`.

OUTPUT:

An iterator.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: class T(tuple):

....: def __le__(left, right):

....: return all(l <= r for l, r in zip(left, right))

sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)),

....: T((4, 4)), T((1, 2)), T((2, 2))])

sage: [(v, type(v)) for v in P.elements_topological()]

[((1, 1), <class '__main__.T'>),

((1, 2), <class '__main__.T'>),

((1, 3), <class '__main__.T'>),

((2, 1), <class '__main__.T'>),

((2, 2), <class '__main__.T'>),

((4, 4), <class '__main__.T'>)]

.. SEEALSO::

:meth:`shells`,

:meth:`shells_topological`,

:meth:`elements`,

:meth:`keys`,

:meth:`keys_topological`,

:meth:`MutablePosetShell.iter_depth_first`,

:meth:`MutablePosetShell.iter_topological`.

"""

for shell in self.shells_topological(**kwargs):

yield shell.element

def keys(self, **kwargs):

r"""

Return an iterator over all keys of the elements.

INPUT:

- ``kwargs`` -- arguments are passed to :meth:`shells`.

OUTPUT:

An iterator.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP([3, 42, 7], key=lambda c: -c)

sage: [(v, type(v)) for v in sorted(P.keys())]

[(-42, <type 'sage.rings.integer.Integer'>),

(-7, <type 'sage.rings.integer.Integer'>),

(-3, <type 'sage.rings.integer.Integer'>)]

sage: [(v, type(v)) for v in sorted(P.elements())]

[(3, <type 'sage.rings.integer.Integer'>),

(7, <type 'sage.rings.integer.Integer'>),

(42, <type 'sage.rings.integer.Integer'>)]

sage: [(v, type(v)) for v in sorted(P.shells(),

....: key=lambda c: c.element)]

[(3, <class 'sage.data_structures.mutable_poset.MutablePosetShell'>),

(7, <class 'sage.data_structures.mutable_poset.MutablePosetShell'>),

(42, <class 'sage.data_structures.mutable_poset.MutablePosetShell'>)]

.. SEEALSO::

:meth:`shells`,

:meth:`shells_topological`,

:meth:`elements`,

:meth:`elements_topological`,

:meth:`keys_topological`,

:meth:`MutablePosetShell.iter_depth_first`,

:meth:`MutablePosetShell.iter_topological`.

"""

for shell in self.shells(**kwargs):

yield shell.key

def keys_topological(self, **kwargs):

r"""

Return an iterator over all keys of the elements in

topological order.

INPUT:

- ``kwargs`` -- arguments are passed to :meth:`shells_topological`.

OUTPUT:

An iterator.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP([(1, 1), (2, 1), (4, 4)],

....: key=lambda c: c[0])

sage: [(v, type(v)) for v in P.keys_topological()]

[(1, <type 'sage.rings.integer.Integer'>),

(2, <type 'sage.rings.integer.Integer'>),

(4, <type 'sage.rings.integer.Integer'>)]

sage: [(v, type(v)) for v in P.elements_topological()]

[((1, 1), <... 'tuple'>),

((2, 1), <... 'tuple'>),

((4, 4), <... 'tuple'>)]

sage: [(v, type(v)) for v in P.shells_topological()]

[((1, 1), <class 'sage.data_structures.mutable_poset.MutablePosetShell'>),

((2, 1), <class 'sage.data_structures.mutable_poset.MutablePosetShell'>),

((4, 4), <class 'sage.data_structures.mutable_poset.MutablePosetShell'>)]

.. SEEALSO::

:meth:`shells`,

:meth:`shells_topological`,

:meth:`elements`,

:meth:`elements_topological`,

:meth:`keys`,

:meth:`MutablePosetShell.iter_depth_first`,

:meth:`MutablePosetShell.iter_topological`.

"""

for shell in self.shells_topological(**kwargs):

yield shell.key

def repr(self, include_special=False, reverse=False):

r"""

Return a representation of the poset.

INPUT:

- ``include_special`` -- (default: ``False``) a boolean

indicating whether to include the special elements

``'null'`` and ``'oo'`` or not.

- ``reverse`` -- (default: ``False``) a boolean. If set, then

largest elements are displayed first.

OUTPUT:

A string.

.. SEEALSO::

:meth:`repr_full`

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: print(MP().repr())

poset()

"""

s = 'poset('

s += ', '.join(repr(shell) for shell in

self.shells_topological(include_special, reverse))

s += ')'

return s

def repr_full(self, reverse=False):

r"""

Return a representation with ordering details of the poset.

INPUT:

- ``reverse`` -- (default: ``False``) a boolean. If set, then

largest elements are displayed first.

OUTPUT:

A string.

.. SEEALSO::

:meth:`repr`

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: print(MP().repr_full(reverse=True))

poset()

+-- oo

| +-- no successors

| +-- predecessors: null

+-- null

| +-- successors: oo

| +-- no predecessors

"""

sortedshells = tuple(

self.shells_topological(include_special=True, reverse=reverse))

strings = [self.repr(include_special=False, reverse=reverse)]

for shell in sortedshells:

strings.append('+-- ' + repr(shell))

for rev in (not reverse, reverse):

what = 'successors' if not rev else 'predecessors'

if shell.successors(rev):

s = '| +-- ' + what + ': '

s += ', '.join(repr(e) for e in

sortedshells if e in shell.successors(rev))

else:

s = '| +-- no ' + what

strings.append(s)

return '\n'.join(strings)

_repr_ = repr

def contains(self, key):

r"""

Test whether ``key`` is encapsulated by one of the poset's elements.

INPUT:

- ``key`` -- an object.

OUTPUT:

``True`` or ``False``.

.. SEEALSO::

:meth:`shells`,

:meth:`elements`,

:meth:`keys`.

TESTS::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: class T(tuple):

....: def __le__(left, right):

....: return all(l <= r for l, r in zip(left, right))

sage: P = MP()

sage: P.add(T((1, 1)))

sage: T((1, 1)) in P # indirect doctest

True

sage: T((1, 2)) in P # indirect doctest

False

"""

return key in self._shells_

__contains__ = contains

def add(self, element):

r"""

Add the given object as element to the poset.

INPUT:

- ``element`` -- an object (hashable and supporting comparison

with the operator ``<=``).

OUTPUT:

Nothing.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: class T(tuple):

....: def __le__(left, right):

....: return all(l <= r for l, r in zip(left, right))

sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)),

....: T((4, 4)), T((1, 2))])

sage: print(P.repr_full(reverse=True))

poset((4, 4), (1, 3), (1, 2), (2, 1), (1, 1))

+-- oo

| +-- no successors

| +-- predecessors: (4, 4)

+-- (4, 4)

| +-- successors: oo

| +-- predecessors: (1, 3), (2, 1)

+-- (1, 3)

| +-- successors: (4, 4)

| +-- predecessors: (1, 2)

+-- (1, 2)

| +-- successors: (1, 3)

| +-- predecessors: (1, 1)

+-- (2, 1)

| +-- successors: (4, 4)

| +-- predecessors: (1, 1)

+-- (1, 1)

| +-- successors: (1, 2), (2, 1)

| +-- predecessors: null

+-- null

| +-- successors: (1, 1)

| +-- no predecessors

sage: P.add(T((2, 2)))

sage: reprP = P.repr_full(reverse=True); print(reprP)

poset((4, 4), (1, 3), (2, 2), (1, 2), (2, 1), (1, 1))

+-- oo

| +-- no successors

| +-- predecessors: (4, 4)

+-- (4, 4)

| +-- successors: oo

| +-- predecessors: (1, 3), (2, 2)

+-- (1, 3)

| +-- successors: (4, 4)

| +-- predecessors: (1, 2)

+-- (2, 2)

| +-- successors: (4, 4)

| +-- predecessors: (1, 2), (2, 1)

+-- (1, 2)

| +-- successors: (1, 3), (2, 2)

| +-- predecessors: (1, 1)

+-- (2, 1)

| +-- successors: (2, 2)

| +-- predecessors: (1, 1)

+-- (1, 1)

| +-- successors: (1, 2), (2, 1)

| +-- predecessors: null

+-- null

| +-- successors: (1, 1)

| +-- no predecessors

When adding an element which is already in the poset, nothing happens::

sage: e = T((2, 2))

sage: P.add(e)

sage: P.repr_full(reverse=True) == reprP

True

We can influence the behavior when an element with existing key

is to be inserted in the poset. For example, we can perform an

addition on some argument of the elements::

sage: def add(left, right):

....: return (left[0], ''.join(sorted(left[1] + right[1])))

sage: A = MP(key=lambda k: k[0], merge=add)

sage: A.add((3, 'a'))

sage: A

poset((3, 'a'))

sage: A.add((3, 'b'))

sage: A

poset((3, 'ab'))

We can also deal with cancellations. If the return value of

our hook-function is ``None``, then the element is removed out of

the poset::

sage: def add_None(left, right):

....: s = left[1] + right[1]

....: if s == 0:

....: return None

....: return (left[0], s)

sage: B = MP(key=lambda k: k[0],

....: merge=add_None)

sage: B.add((7, 42))

sage: B.add((7, -42))

sage: B

poset()

.. SEEALSO::

:meth:`discard`,

:meth:`pop`,

:meth:`remove`.

TESTS::

sage: R = MP([(1, 1, 42), (1, 3, 42), (2, 1, 7),

....: (4, 4, 42), (1, 2, 7), (2, 2, 7)],

....: key=lambda k: T(k[2:3]))

sage: print(R.repr_full(reverse=True))

poset((1, 1, 42), (2, 1, 7))

+-- oo

| +-- no successors

| +-- predecessors: (1, 1, 42)

+-- (1, 1, 42)

| +-- successors: oo

| +-- predecessors: (2, 1, 7)

+-- (2, 1, 7)

| +-- successors: (1, 1, 42)

| +-- predecessors: null

+-- null

| +-- successors: (2, 1, 7)

| +-- no predecessors

sage: P = MP()

sage: P.add(None)

Traceback (most recent call last):

...

ValueError: None is not an allowed element.

"""

if element is None:

raise ValueError('None is not an allowed element.')

key = self.get_key(element)

if key in self._shells_:

if self._merge_ is not None:

self.shell(key).merge(element, delete=False)

return

new = MutablePosetShell(self, element)

new._predecessors_ = self.null.lower_covers(new)

new._successors_ = self.oo.upper_covers(new)

for s in new.predecessors():

for l in s.successors().intersection(new.successors()):

l.predecessors().remove(s)

s.successors().remove(l)

s.successors().add(new)

for l in new.successors():

l.predecessors().add(new)

self._shells_[key] = new

def remove(self, key, raise_key_error=True):

r"""

Remove the given object from the poset.

INPUT:

- ``key`` -- the key of an object.

- ``raise_key_error`` -- (default: ``True``) switch raising

``KeyError`` on and off.

OUTPUT:

Nothing.

If the element is not a member and ``raise_key_error`` is set

(default), raise a ``KeyError``.

.. NOTE::

As with Python's ``set``, the methods :meth:`remove`

and :meth:`discard` only differ in their behavior when an

element is not contained in the poset: :meth:`remove`

raises a ``KeyError`` whereas :meth:`discard` does not

raise any exception.

This default behavior can be overridden with the

``raise_key_error`` parameter.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: class T(tuple):

....: def __le__(left, right):

....: return all(l <= r for l, r in zip(left, right))

sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)),

....: T((4, 4)), T((1, 2)), T((2, 2))])

sage: print(P.repr_full(reverse=True))

poset((4, 4), (1, 3), (2, 2), (1, 2), (2, 1), (1, 1))

+-- oo

| +-- no successors

| +-- predecessors: (4, 4)

+-- (4, 4)

| +-- successors: oo

| +-- predecessors: (1, 3), (2, 2)

+-- (1, 3)

| +-- successors: (4, 4)

| +-- predecessors: (1, 2)

+-- (2, 2)

| +-- successors: (4, 4)

| +-- predecessors: (1, 2), (2, 1)

+-- (1, 2)

| +-- successors: (1, 3), (2, 2)

| +-- predecessors: (1, 1)

+-- (2, 1)

| +-- successors: (2, 2)

| +-- predecessors: (1, 1)

+-- (1, 1)

| +-- successors: (1, 2), (2, 1)

| +-- predecessors: null

+-- null

| +-- successors: (1, 1)

| +-- no predecessors

sage: P.remove(T((1, 2)))

sage: print(P.repr_full(reverse=True))

poset((4, 4), (1, 3), (2, 2), (2, 1), (1, 1))

+-- oo

| +-- no successors

| +-- predecessors: (4, 4)

+-- (4, 4)

| +-- successors: oo

| +-- predecessors: (1, 3), (2, 2)

+-- (1, 3)

| +-- successors: (4, 4)

| +-- predecessors: (1, 1)

+-- (2, 2)

| +-- successors: (4, 4)

| +-- predecessors: (2, 1)

+-- (2, 1)

| +-- successors: (2, 2)

| +-- predecessors: (1, 1)

+-- (1, 1)

| +-- successors: (1, 3), (2, 1)

| +-- predecessors: null

+-- null

| +-- successors: (1, 1)

| +-- no predecessors

.. SEEALSO::

:meth:`add`,

:meth:`clear`,

:meth:`discard`,

:meth:`pop`.

TESTS::

sage: Q = MP([(1, 1, 42), (1, 3, 42), (2, 1, 7),

....: (4, 4, 42), (1, 2, 7), (2, 2, 7)],

....: key=lambda k: T(k[0:2]))

sage: print(Q.repr_full(reverse=True))

poset((4, 4, 42), (1, 3, 42), (2, 2, 7),

(1, 2, 7), (2, 1, 7), (1, 1, 42))

+-- oo

| +-- no successors

| +-- predecessors: (4, 4, 42)

+-- (4, 4, 42)

| +-- successors: oo

| +-- predecessors: (1, 3, 42), (2, 2, 7)

+-- (1, 3, 42)

| +-- successors: (4, 4, 42)

| +-- predecessors: (1, 2, 7)

+-- (2, 2, 7)

| +-- successors: (4, 4, 42)

| +-- predecessors: (1, 2, 7), (2, 1, 7)

+-- (1, 2, 7)

| +-- successors: (1, 3, 42), (2, 2, 7)

| +-- predecessors: (1, 1, 42)

+-- (2, 1, 7)

| +-- successors: (2, 2, 7)

| +-- predecessors: (1, 1, 42)

+-- (1, 1, 42)

| +-- successors: (1, 2, 7), (2, 1, 7)

| +-- predecessors: null

+-- null

| +-- successors: (1, 1, 42)

| +-- no predecessors

sage: Q.remove((1,1))

sage: print(Q.repr_full(reverse=True))

poset((4, 4, 42), (1, 3, 42), (2, 2, 7), (1, 2, 7), (2, 1, 7))

+-- oo

| +-- no successors

| +-- predecessors: (4, 4, 42)

+-- (4, 4, 42)

| +-- successors: oo

| +-- predecessors: (1, 3, 42), (2, 2, 7)

+-- (1, 3, 42)

| +-- successors: (4, 4, 42)

| +-- predecessors: (1, 2, 7)

+-- (2, 2, 7)

| +-- successors: (4, 4, 42)

| +-- predecessors: (1, 2, 7), (2, 1, 7)

+-- (1, 2, 7)

| +-- successors: (1, 3, 42), (2, 2, 7)

| +-- predecessors: null

+-- (2, 1, 7)

| +-- successors: (2, 2, 7)

| +-- predecessors: null

+-- null

| +-- successors: (1, 2, 7), (2, 1, 7)

| +-- no predecessors

sage: P = MP()

sage: P.remove(None)

Traceback (most recent call last):

...

ValueError: None is not an allowed key.

"""

if key is None:

raise ValueError('None is not an allowed key.')

try:

shell = self._shells_[key]

except KeyError:

if not raise_key_error:

return

raise KeyError('Key %s is not contained in this poset.' % (key,))

for reverse in (False, True):

for p in shell.predecessors(reverse):

S = p.successors(reverse)

S.remove(shell)

D = set(s for s in p.iter_depth_first(reverse)

if s in shell.successors(reverse))

S.update(shell.successors(reverse))

S.difference_update(D)

del self._shells_[key]

def discard(self, key, raise_key_error=False):

r"""

Remove the given object from the poset.

INPUT:

- ``key`` -- the key of an object.

- ``raise_key_error`` -- (default: ``False``) switch raising

``KeyError`` on and off.

OUTPUT:

Nothing.

If the element is not a member and ``raise_key_error`` is set

(not default), raise a ``KeyError``.

.. NOTE::

As with Python's ``set``, the methods :meth:`remove`

and :meth:`discard` only differ in their behavior when an

element is not contained in the poset: :meth:`remove`

raises a ``KeyError`` whereas :meth:`discard` does not

raise any exception.

This default behavior can be overridden with the

``raise_key_error`` parameter.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: class T(tuple):

....: def __le__(left, right):

....: return all(l <= r for l, r in zip(left, right))

sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)),

....: T((4, 4)), T((1, 2)), T((2, 2))])

sage: P.discard(T((1, 2)))

sage: P.remove(T((1, 2)))

Traceback (most recent call last):

...

KeyError: 'Key (1, 2) is not contained in this poset.'

sage: P.discard(T((1, 2)))

.. SEEALSO::

:meth:`add`,

:meth:`clear`,

:meth:`remove`,

:meth:`pop`.

"""

return self.remove(key, raise_key_error)

def pop(self, **kwargs):

r"""

Remove and return an arbitrary poset element.

INPUT:

- ``kwargs`` -- arguments are passed to :meth:`shells_topological`.

OUTPUT:

An object.

.. NOTE::

The special elements ``'null'`` and ``'oo'`` cannot be popped.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP()

sage: P.add(3)

sage: P

poset(3)

sage: P.pop()

sage: P

poset()

sage: P.pop()

Traceback (most recent call last):

...

KeyError: 'pop from an empty poset'

.. SEEALSO::

:meth:`add`,

:meth:`clear`,

:meth:`discard`,

:meth:`remove`.

"""

kwargs['include_special'] = False

try:

shell = next(self.shells_topological(**kwargs))

except StopIteration:

raise KeyError('pop from an empty poset')

self.remove(shell.key)

return shell.element

def union(self, *other):

r"""

Return the union of the given posets as a new poset

INPUT:

- ``other`` -- a poset or an iterable. In the latter case the

iterated objects are seen as elements of a poset.

It is possible to specify more than one ``other`` as

variadic arguments (arbitrary argument lists).

OUTPUT:

A poset.

.. NOTE::

The key of an element is used for comparison. Thus elements with

the same key are considered as equal.

Due to keys and a ``merge`` function (see :class:`MutablePoset`)

this operation might not be commutative.

.. TODO::

Use the already existing information in the other poset to speed

up this function. (At the moment each element of the other poset

is inserted one by one and without using this information.)

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP([3, 42, 7]); P

poset(3, 7, 42)

sage: Q = MP([4, 8, 42]); Q

poset(4, 8, 42)

sage: P.union(Q)

poset(3, 4, 7, 8, 42)

.. SEEALSO::

:meth:`union_update`,

:meth:`difference`, :meth:`difference_update`,

:meth:`intersection`, :meth:`intersection_update`,

:meth:`symmetric_difference`, :meth:`symmetric_difference_update`,

:meth:`is_disjoint`,

:meth:`is_subset`,

:meth:`is_superset`.

TESTS::

sage: P.union(P, Q, Q, P)

poset(3, 4, 7, 8, 42)

"""

new = self.copy()

new.update(*other)

return new

def union_update(self, *other):

r"""

Update this poset with the union of itself and another poset.

INPUT:

- ``other`` -- a poset or an iterable. In the latter case the

iterated objects are seen as elements of a poset.

It is possible to specify more than one ``other`` as

variadic arguments (arbitrary argument lists).

OUTPUT:

Nothing.

.. NOTE::

The key of an element is used for comparison. Thus elements with

the same key are considered as equal;

``A.union_update(B)`` and ``B.union_update(A)`` might

result in different posets.

.. TODO::

Use the already existing information in the other poset to speed

up this function. (At the moment each element of the other poset

is inserted one by one and without using this information.)

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP([3, 42, 7]); P

poset(3, 7, 42)

sage: Q = MP([4, 8, 42]); Q

poset(4, 8, 42)

sage: P.union_update(Q)

sage: P

poset(3, 4, 7, 8, 42)

.. SEEALSO::

:meth:`union`,

:meth:`difference`, :meth:`difference_update`,

:meth:`intersection`, :meth:`intersection_update`,

:meth:`symmetric_difference`, :meth:`symmetric_difference_update`,

:meth:`is_disjoint`,

:meth:`is_subset`,

:meth:`is_superset`.

TESTS::

sage: Q.update(P)

sage: Q

poset(3, 4, 7, 8, 42)

"""

for o in other:

try:

it = o.elements()

except AttributeError:

it = iter(o)

for element in it:

self.add(element)

update = union_update # as in a Python set

r"""

Alias of :meth:`union_update`.

"""

def difference(self, *other):

r"""

Return a new poset where all elements of this poset, which are

contained in one of the other given posets, are removed.

INPUT:

- ``other`` -- a poset or an iterable. In the latter case the

iterated objects are seen as elements of a poset.

It is possible to specify more than one ``other`` as

variadic arguments (arbitrary argument lists).

OUTPUT:

A poset.

.. NOTE::

The key of an element is used for comparison. Thus elements with

the same key are considered as equal.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP([3, 42, 7]); P

poset(3, 7, 42)

sage: Q = MP([4, 8, 42]); Q

poset(4, 8, 42)

sage: P.difference(Q)

poset(3, 7)

.. SEEALSO::

:meth:`union`, :meth:`union_update`,

:meth:`difference_update`,

:meth:`intersection`, :meth:`intersection_update`,

:meth:`symmetric_difference`, :meth:`symmetric_difference_update`,

:meth:`is_disjoint`,

:meth:`is_subset`,

:meth:`is_superset`.

TESTS::

sage: P.difference(Q, Q)

poset(3, 7)

sage: P.difference(P)

poset()

sage: P.difference(Q, P)

poset()

"""

new = self.copy()

new.difference_update(*other)

return new

def difference_update(self, *other):

r"""

Remove all elements of another poset from this poset.

INPUT:

- ``other`` -- a poset or an iterable. In the latter case the

iterated objects are seen as elements of a poset.

It is possible to specify more than one ``other`` as

variadic arguments (arbitrary argument lists).

OUTPUT:

Nothing.

.. NOTE::

The key of an element is used for comparison. Thus elements with

the same key are considered as equal.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP([3, 42, 7]); P

poset(3, 7, 42)

sage: Q = MP([4, 8, 42]); Q

poset(4, 8, 42)

sage: P.difference_update(Q)

sage: P

poset(3, 7)

.. SEEALSO::

:meth:`union`, :meth:`union_update`,

:meth:`difference`,

:meth:`intersection`, :meth:`intersection_update`,

:meth:`symmetric_difference`, :meth:`symmetric_difference_update`,

:meth:`is_disjoint`,

:meth:`is_subset`,

:meth:`is_superset`.

"""

for o in other:

try:

it = o.keys()

except AttributeError:

it = iter(o)

for key in it:

self.discard(key)

def intersection(self, *other):

r"""

Return the intersection of the given posets as a new poset

INPUT:

- ``other`` -- a poset or an iterable. In the latter case the

iterated objects are seen as elements of a poset.

It is possible to specify more than one ``other`` as

variadic arguments (arbitrary argument lists).

OUTPUT:

A poset.

.. NOTE::

The key of an element is used for comparison. Thus elements with

the same key are considered as equal.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP([3, 42, 7]); P

poset(3, 7, 42)

sage: Q = MP([4, 8, 42]); Q

poset(4, 8, 42)

sage: P.intersection(Q)

poset(42)

.. SEEALSO::

:meth:`union`, :meth:`union_update`,

:meth:`difference`, :meth:`difference_update`,

:meth:`intersection_update`,

:meth:`symmetric_difference`, :meth:`symmetric_difference_update`,

:meth:`is_disjoint`,

:meth:`is_subset`,

:meth:`is_superset`.

TESTS::

sage: P.intersection(P, Q, Q, P)

poset(42)

"""

new = self.copy()

new.intersection_update(*other)

return new

def intersection_update(self, *other):

r"""

Update this poset with the intersection of itself and another poset.

INPUT:

- ``other`` -- a poset or an iterable. In the latter case the

iterated objects are seen as elements of a poset.

It is possible to specify more than one ``other`` as

variadic arguments (arbitrary argument lists).

OUTPUT:

Nothing.

.. NOTE::

The key of an element is used for comparison. Thus elements with

the same key are considered as equal;

``A.intersection_update(B)`` and ``B.intersection_update(A)`` might

result in different posets.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP([3, 42, 7]); P

poset(3, 7, 42)

sage: Q = MP([4, 8, 42]); Q

poset(4, 8, 42)

sage: P.intersection_update(Q)

sage: P

poset(42)

.. SEEALSO::

:meth:`union`, :meth:`union_update`,

:meth:`difference`, :meth:`difference_update`,

:meth:`intersection`,

:meth:`symmetric_difference`, :meth:`symmetric_difference_update`,

:meth:`is_disjoint`,

:meth:`is_subset`,

:meth:`is_superset`.

"""

keys = tuple(self.keys())

for key in keys:

if any(key not in o for o in other):

self.discard(key)

def symmetric_difference(self, other):

r"""

Return the symmetric difference of two posets as a new poset.

INPUT:

- ``other`` -- a poset.

OUTPUT:

A poset.

.. NOTE::

The key of an element is used for comparison. Thus elements with

the same key are considered as equal.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP([3, 42, 7]); P

poset(3, 7, 42)

sage: Q = MP([4, 8, 42]); Q

poset(4, 8, 42)

sage: P.symmetric_difference(Q)

poset(3, 4, 7, 8)

.. SEEALSO::

:meth:`union`, :meth:`union_update`,

:meth:`difference`, :meth:`difference_update`,

:meth:`intersection`, :meth:`intersection_update`,

:meth:`symmetric_difference_update`,

:meth:`is_disjoint`,

:meth:`is_subset`,

:meth:`is_superset`.

"""

new = self.copy()

new.symmetric_difference_update(other)

return new

def symmetric_difference_update(self, other):

r"""

Update this poset with the symmetric difference of itself and

another poset.

INPUT:

- ``other`` -- a poset.

OUTPUT:

Nothing.

.. NOTE::

The key of an element is used for comparison. Thus elements with

the same key are considered as equal;

``A.symmetric_difference_update(B)`` and

``B.symmetric_difference_update(A)`` might

result in different posets.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP([3, 42, 7]); P

poset(3, 7, 42)

sage: Q = MP([4, 8, 42]); Q

poset(4, 8, 42)

sage: P.symmetric_difference_update(Q)

sage: P

poset(3, 4, 7, 8)

.. SEEALSO::

:meth:`union`, :meth:`union_update`,

:meth:`difference`, :meth:`difference_update`,

:meth:`intersection`, :meth:`intersection_update`,

:meth:`symmetric_difference`,

:meth:`is_disjoint`,

:meth:`is_subset`,

:meth:`is_superset`.

"""

T = other.difference(self)

self.difference_update(other)

self.union_update(T)

def is_disjoint(self, other):

r"""

Return whether another poset is disjoint to this poset.

INPUT:

- ``other`` -- a poset or an iterable. In the latter case the

iterated objects are seen as elements of a poset.

OUTPUT:

Nothing.

.. NOTE::

If this poset uses a ``key``-function, then all

comparisons are performed on the keys of the elements (and

not on the elements themselves).

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP([3, 42, 7]); P

poset(3, 7, 42)

sage: Q = MP([4, 8, 42]); Q

poset(4, 8, 42)

sage: P.is_disjoint(Q)

False

sage: P.is_disjoint(Q.difference(P))

True

.. SEEALSO::

:meth:`is_subset`,

:meth:`is_superset`,

:meth:`union`, :meth:`union_update`,

:meth:`difference`, :meth:`difference_update`,

:meth:`intersection`, :meth:`intersection_update`,

:meth:`symmetric_difference`, :meth:`symmetric_difference_update`.

"""

return all(key not in other for key in self.keys())

isdisjoint = is_disjoint # as in a Python set

r"""

Alias of :meth:`is_disjoint`.

"""

def is_subset(self, other):

r"""

Return whether another poset contains this poset, i.e., whether this poset

is a subset of the other poset.

INPUT:

- ``other`` -- a poset or an iterable. In the latter case the

iterated objects are seen as elements of a poset.

OUTPUT:

Nothing.

.. NOTE::

If this poset uses a ``key``-function, then all

comparisons are performed on the keys of the elements (and

not on the elements themselves).

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP([3, 42, 7]); P

poset(3, 7, 42)

sage: Q = MP([4, 8, 42]); Q

poset(4, 8, 42)

sage: P.is_subset(Q)

False

sage: Q.is_subset(P)

False

sage: P.is_subset(P)

True

sage: P.is_subset(P.union(Q))

True

.. SEEALSO::

:meth:`is_disjoint`,

:meth:`is_superset`,

:meth:`union`, :meth:`union_update`,

:meth:`difference`, :meth:`difference_update`,

:meth:`intersection`, :meth:`intersection_update`,

:meth:`symmetric_difference`, :meth:`symmetric_difference_update`.

"""

return all(key in other for key in self.keys())

issubset = is_subset # as in a Python set

r"""

Alias of :meth:`is_subset`.

"""

def is_superset(self, other):

r"""

Return whether this poset contains another poset, i.e., whether this poset

is a superset of the other poset.

INPUT:

- ``other`` -- a poset or an iterable. In the latter case the

iterated objects are seen as elements of a poset.

OUTPUT:

Nothing.

.. NOTE::

If this poset uses a ``key``-function, then all

comparisons are performed on the keys of the elements (and

not on the elements themselves).

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: P = MP([3, 42, 7]); P

poset(3, 7, 42)

sage: Q = MP([4, 8, 42]); Q

poset(4, 8, 42)

sage: P.is_superset(Q)

False

sage: Q.is_superset(P)

False

sage: P.is_superset(P)

True

sage: P.union(Q).is_superset(P)

True

.. SEEALSO::

:meth:`is_disjoint`,

:meth:`is_subset`,

:meth:`union`, :meth:`union_update`,

:meth:`difference`, :meth:`difference_update`,

:meth:`intersection`, :meth:`intersection_update`,

:meth:`symmetric_difference`, :meth:`symmetric_difference_update`.

"""

try:

it = other.keys()

except AttributeError:

it = iter(other)

return all(key in self for key in it)

issuperset = is_superset # as in a Python set

r"""

Alias of :meth:`is_superset`.

"""

def merge(self, key=None, reverse=False):

r"""

Merge the given element with its successors/predecessors.

INPUT:

- ``key`` -- the key specifying an element or ``None``

(default), in which case this method is called on each

element in this poset.

- ``reverse`` -- (default: ``False``) specifies which

direction to go first:

``False`` searches towards ``'oo'`` and

``True`` searches towards ``'null'``.

When ``key=None``, then this also

specifies which elements are merged first.

OUTPUT:

Nothing.

This method tests all (not necessarily direct) successors and

predecessors of the given element whether they can be merged with

the element itself. This is done by the ``can_merge``-function

of :class:`MutablePoset`. If this merge is possible, then it

is performed by calling :class:`MutablePoset`'s

``merge``-function and the corresponding successor/predecessor

is removed from the poset.

.. NOTE::

``can_merge`` is applied in the sense of the condition of

depth first iteration, i.e., once ``can_merge`` fails,

the successors/predecessors are no longer tested.

.. NOTE::

The motivation for such a merge behavior comes from

asymptotic expansions: `O(n^3)` merges with, for

example, `3n^2` or `O(n)` to `O(n^3)` (as `n` tends to

`\infty`; see :wikipedia:`Big_O_notation`).

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: class T(tuple):

....: def __le__(left, right):

....: return all(l <= r for l, r in zip(left, right))

sage: key = lambda t: T(t[0:2])

sage: def add(left, right):

....: return (left[0], left[1],

....: ''.join(sorted(left[2] + right[2])))

sage: def can_add(left, right):

....: return key(left) >= key(right)

sage: P = MP([(1, 1, 'a'), (1, 3, 'b'), (2, 1, 'c'),

....: (4, 4, 'd'), (1, 2, 'e'), (2, 2, 'f')],

....: key=key, merge=add, can_merge=can_add)

sage: Q = copy(P)

sage: Q.merge(T((1, 3)))

sage: print(Q.repr_full(reverse=True))

poset((4, 4, 'd'), (1, 3, 'abe'), (2, 2, 'f'), (2, 1, 'c'))

+-- oo

| +-- no successors

| +-- predecessors: (4, 4, 'd')

+-- (4, 4, 'd')

| +-- successors: oo

| +-- predecessors: (1, 3, 'abe'), (2, 2, 'f')

+-- (1, 3, 'abe')

| +-- successors: (4, 4, 'd')

| +-- predecessors: null

+-- (2, 2, 'f')

| +-- successors: (4, 4, 'd')

| +-- predecessors: (2, 1, 'c')

+-- (2, 1, 'c')

| +-- successors: (2, 2, 'f')

| +-- predecessors: null

+-- null

| +-- successors: (1, 3, 'abe'), (2, 1, 'c')

| +-- no predecessors

sage: for k in P.keys():

....: Q = copy(P)

....: Q.merge(k)

....: print('merging %s: %s' % (k, Q))

merging (1, 2): poset((1, 2, 'ae'), (1, 3, 'b'),

(2, 1, 'c'), (2, 2, 'f'), (4, 4, 'd'))

merging (1, 3): poset((1, 3, 'abe'), (2, 1, 'c'),

(2, 2, 'f'), (4, 4, 'd'))

merging (4, 4): poset((4, 4, 'abcdef'))

merging (2, 1): poset((1, 2, 'e'), (1, 3, 'b'),

(2, 1, 'ac'), (2, 2, 'f'), (4, 4, 'd'))

merging (2, 2): poset((1, 3, 'b'), (2, 2, 'acef'), (4, 4, 'd'))

merging (1, 1): poset((1, 1, 'a'), (1, 2, 'e'), (1, 3, 'b'),

(2, 1, 'c'), (2, 2, 'f'), (4, 4, 'd'))

sage: Q = copy(P)

sage: Q.merge(); Q

poset((4, 4, 'abcdef'))

.. SEEALSO::

:meth:`MutablePosetShell.merge`

TESTS::

sage: copy(P).merge(reverse=False) == copy(P).merge(reverse=True)

True

sage: P = MP(srange(4),

....: merge=lambda l, r: l, can_merge=lambda l, r: l >= r); P

poset(0, 1, 2, 3)

sage: Q = P.copy()

sage: Q.merge(reverse=True); Q

poset(3)

sage: R = P.mapped(lambda x: x+1)

sage: R.merge(reverse=True); R

poset(4)

sage: P = MP(srange(4),

....: merge=lambda l, r: r, can_merge=lambda l, r: l < r)

sage: P.merge()

Traceback (most recent call last):

...

RuntimeError: Stopping merge before started;

the can_merge-function is not reflexive.

"""

if key is None:

for shell in tuple(self.shells_topological(reverse=reverse)):

if shell.key in self._shells_:

self.merge(key=shell.key)

return

shell = self.shell(key)

def can_merge(other):

return self._can_merge_(shell.element, other.element)

for rev in (reverse, not reverse):

to_merge = shell.iter_depth_first(

reverse=rev, condition=can_merge)

try:

next(to_merge)

except StopIteration:

raise RuntimeError('Stopping merge before started; the '

'can_merge-function is not reflexive.')

for m in tuple(to_merge):

if m.is_special():

continue

shell.merge(m.element, check=False, delete=True)

def maximal_elements(self):

r"""

Return an iterator over the maximal elements of this poset.

INPUT:

Nothing.

OUTPUT:

An iterator.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: class T(tuple):

....: def __le__(left, right):

....: return all(l <= r for l, r in zip(left, right))

sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)),

....: T((1, 2)), T((2, 2))])

sage: list(P.maximal_elements())

[(1, 3), (2, 2)]

.. SEEALSO::

:meth:`minimal_elements`

"""

return iter(shell.element

for shell in self.oo.predecessors()

if not shell.is_special())

def minimal_elements(self):

r"""

Return an iterator over the minimal elements of this poset.

INPUT:

Nothing.

OUTPUT:

An iterator.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: class T(tuple):

....: def __le__(left, right):

....: return all(l <= r for l, r in zip(left, right))

sage: P = MP([T((1, 3)), T((2, 1)),

....: T((4, 4)), T((1, 2)), T((2, 2))])

sage: list(P.minimal_elements())

[(1, 2), (2, 1)]

.. SEEALSO::

:meth:`maximal_elements`

"""

return iter(shell.element

for shell in self.null.successors()

if not shell.is_special())

def map(self, function, topological=False, reverse=False):

r"""

Apply the given ``function`` to each element of this poset.

INPUT:

- ``function`` -- a function mapping an existing element to

a new element.

- ``topological`` -- (default: ``False``) if set, then the

mapping is done in topological order, otherwise unordered.

- ``reverse`` -- is passed on to topological ordering.

OUTPUT:

Nothing.

.. NOTE::

Since this method works inplace, it is not allowed that

``function`` alters the key of an element.

.. NOTE::

If ``function`` returns ``None``, then the element is

removed.

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: class T(tuple):

....: def __le__(left, right):

....: return all(l <= r for l, r in zip(left, right))

sage: P = MP([T((1, 3)), T((2, 1)),

....: T((4, 4)), T((1, 2)), T((2, 2))],

....: key=lambda e: e[:2])

sage: P.map(lambda e: e + (sum(e),))

sage: P

poset((1, 2, 3), (1, 3, 4), (2, 1, 3), (2, 2, 4), (4, 4, 8))

TESTS::

sage: P.map(lambda e: e if e[2] != 4 else None); P

poset((1, 2, 3), (2, 1, 3), (4, 4, 8))

.. SEEALSO::

:meth:`copy`,

:meth:`mapped`.

"""

shells = self.shells_topological(reverse=reverse) \

if topological else self.shells()

remove = []

for shell in shells:

shell._element_ = function(shell._element_)

if shell._element_ is None:

remove.append(shell.key)

for key in remove:

self.remove(key)

def mapped(self, function):

r"""

Return a poset where on each element the given ``function``

was applied.

INPUT:

- ``function`` -- a function mapping an existing element to

a new element.

- ``topological`` -- (default: ``False``) if set, then the

mapping is done in topological order, otherwise unordered.

- ``reverse`` -- is passed on to topological ordering.

OUTPUT:

A :class:`MutablePoset`.

.. NOTE::

``function`` is not allowed to change the order of the keys,

but changing the keys themselves is allowed (in contrast

to :meth:`map`).

EXAMPLES::

sage: from sage.data_structures.mutable_poset import MutablePoset as MP

sage: class T(tuple):

....: def __le__(left, right):

....: return all(l <= r for l, r in zip(left, right))

sage: P = MP([T((1, 3)), T((2, 1)),

....: T((4, 4)), T((1, 2)), T((2, 2))])

sage: P.mapped(lambda e: str(e))

poset('(1, 2)', '(1, 3)', '(2, 1)', '(2, 2)', '(4, 4)')

.. SEEALSO::

:meth:`copy`,

:meth:`map`.

"""

return self.copy(mapping=function)

# *****************************************************************************

Coverage for local/lib/python2.7/site-packages/sage/data_structures/mutable_poset.py : 98%

363 statements 357 run 6 missing 0 excluded