Data structures for maps between finite sets¶
This module implements several fast Cython data structures for maps
between two finite set. Those classes are not intended to be used
directly. Instead, such a map should be constructed via its parent,
using the class FiniteSetMaps.
EXAMPLES:
To create a map between two sets, one first creates the set of such maps:
sage: M = FiniteSetMaps(["a", "b"], [3, 4, 5])
>>> from sage.all import *
>>> M = FiniteSetMaps(["a", "b"], [Integer(3), Integer(4), Integer(5)])
The map can then be constructed either from a function:
sage: f1 = M(lambda c: ord(c)-94); f1
map: a -> 3, b -> 4
>>> from sage.all import *
>>> f1 = M(lambda c: ord(c)-Integer(94)); f1
map: a -> 3, b -> 4
or from a dictionary:
sage: f2 = M.from_dict({'a':3, 'b':4}); f2
map: a -> 3, b -> 4
>>> from sage.all import *
>>> f2 = M.from_dict({'a':Integer(3), 'b':Integer(4)}); f2
map: a -> 3, b -> 4
The two created maps are equal:
sage: f1 == f2
True
>>> from sage.all import *
>>> f1 == f2
True
Internally, maps are represented as the list of the ranks of the
images f(x) in the co-domain, in the order of the domain:
sage: list(f2)
[0, 1]
>>> from sage.all import *
>>> list(f2)
[0, 1]
A third fast way to create a map it to use such a list. it should be kept for internal use:
sage: f3 = M._from_list_([0, 1]); f3
map: a -> 3, b -> 4
sage: f1 == f3
True
>>> from sage.all import *
>>> f3 = M._from_list_([Integer(0), Integer(1)]); f3
map: a -> 3, b -> 4
>>> f1 == f3
True
AUTHORS:
Florent Hivert
- class sage.sets.finite_set_map_cy.FiniteSetEndoMap_N[source]¶
Bases:
FiniteSetMap_MNMap from
range(n)to itself.See also
FiniteSetMap_MNfor assumptions on the parent
- class sage.sets.finite_set_map_cy.FiniteSetEndoMap_Set[source]¶
Bases:
FiniteSetMap_SetMap from a set to itself.
See also
FiniteSetMap_Setfor assumptions on the parent
- class sage.sets.finite_set_map_cy.FiniteSetMap_MN[source]¶
Bases:
ClonableIntArrayData structure for maps from
range(m)torange(n).We assume that the parent given as argument is such that:
m– stored inself.parent()._mn– stored inself.parent()._nthe domain is in
self.parent().domain()the codomain is in
self.parent().codomain()
- check()[source]¶
Perform checks on
self.Check that
selfis a proper function and then callsparent.check_element(self)whereparentis the parent ofself.
- codomain()[source]¶
Return the codomain of
self.EXAMPLES:
sage: FiniteSetMaps(4, 3)([1, 0, 2, 1]).codomain() {0, 1, 2}
>>> from sage.all import * >>> FiniteSetMaps(Integer(4), Integer(3))([Integer(1), Integer(0), Integer(2), Integer(1)]).codomain() {0, 1, 2}
- domain()[source]¶
Return the domain of
self.EXAMPLES:
sage: FiniteSetMaps(4, 3)([1, 0, 2, 1]).domain() {0, 1, 2, 3}
>>> from sage.all import * >>> FiniteSetMaps(Integer(4), Integer(3))([Integer(1), Integer(0), Integer(2), Integer(1)]).domain() {0, 1, 2, 3}
- fibers()[source]¶
Return the fibers of
self.OUTPUT:
a dictionary
dsuch thatd[y]is the set of allxindomainsuch thatf(x) = yEXAMPLES:
sage: FiniteSetMaps(4, 3)([1, 0, 2, 1]).fibers() {0: {1}, 1: {0, 3}, 2: {2}} sage: F = FiniteSetMaps(["a", "b", "c"]) sage: F.from_dict({"a": "b", "b": "a", "c": "b"}).fibers() == {'a': {'b'}, 'b': {'a', 'c'}} True
>>> from sage.all import * >>> FiniteSetMaps(Integer(4), Integer(3))([Integer(1), Integer(0), Integer(2), Integer(1)]).fibers() {0: {1}, 1: {0, 3}, 2: {2}} >>> F = FiniteSetMaps(["a", "b", "c"]) >>> F.from_dict({"a": "b", "b": "a", "c": "b"}).fibers() == {'a': {'b'}, 'b': {'a', 'c'}} True
- getimage(i)[source]¶
Return the image of
ibyself.INPUT:
i– any object
Note
if you need speed, please use instead
_getimage()EXAMPLES:
sage: fs = FiniteSetMaps(4, 3)([1, 0, 2, 1]) sage: fs.getimage(0), fs.getimage(1), fs.getimage(2), fs.getimage(3) (1, 0, 2, 1)
>>> from sage.all import * >>> fs = FiniteSetMaps(Integer(4), Integer(3))([Integer(1), Integer(0), Integer(2), Integer(1)]) >>> fs.getimage(Integer(0)), fs.getimage(Integer(1)), fs.getimage(Integer(2)), fs.getimage(Integer(3)) (1, 0, 2, 1)
- image_set()[source]¶
Return the image set of
self.EXAMPLES:
sage: FiniteSetMaps(4, 3)([1, 0, 2, 1]).image_set() {0, 1, 2} sage: FiniteSetMaps(4, 3)([1, 0, 0, 1]).image_set() {0, 1}
>>> from sage.all import * >>> FiniteSetMaps(Integer(4), Integer(3))([Integer(1), Integer(0), Integer(2), Integer(1)]).image_set() {0, 1, 2} >>> FiniteSetMaps(Integer(4), Integer(3))([Integer(1), Integer(0), Integer(0), Integer(1)]).image_set() {0, 1}
- items()[source]¶
The items of
self.Return the list of the ordered pairs
(x, self(x))EXAMPLES:
sage: FiniteSetMaps(4, 3)([1, 0, 2, 1]).items() [(0, 1), (1, 0), (2, 2), (3, 1)]
>>> from sage.all import * >>> FiniteSetMaps(Integer(4), Integer(3))([Integer(1), Integer(0), Integer(2), Integer(1)]).items() [(0, 1), (1, 0), (2, 2), (3, 1)]
- setimage(i, j)[source]¶
Set the image of
iasjinself.Warning
selfmust be mutable; otherwise an exception is raised.INPUT:
i,j– twoobject’s
OUTPUT: none
Note
if you need speed, please use instead
_setimage()EXAMPLES:
sage: fs = FiniteSetMaps(4, 3)([1, 0, 2, 1]) sage: fs2 = copy(fs) sage: fs2.setimage(2, 1) sage: fs2 [1, 0, 1, 1] sage: with fs.clone() as fs3: ....: fs3.setimage(0, 2) ....: fs3.setimage(1, 2) sage: fs3 [2, 2, 2, 1]
>>> from sage.all import * >>> fs = FiniteSetMaps(Integer(4), Integer(3))([Integer(1), Integer(0), Integer(2), Integer(1)]) >>> fs2 = copy(fs) >>> fs2.setimage(Integer(2), Integer(1)) >>> fs2 [1, 0, 1, 1] >>> with fs.clone() as fs3: ... fs3.setimage(Integer(0), Integer(2)) ... fs3.setimage(Integer(1), Integer(2)) >>> fs3 [2, 2, 2, 1]
- class sage.sets.finite_set_map_cy.FiniteSetMap_Set[source]¶
Bases:
FiniteSetMap_MNData structure for maps.
We assume that the parent given as argument is such that:
the domain is in
parent.domain()the codomain is in
parent.codomain()parent._mcontains the cardinality of the domainparent._ncontains the cardinality of the codomainparent._unrank_domainandparent._rank_domainis a pair of reciprocal rank and unrank functions between the domain andrange(parent._m).parent._unrank_codomainandparent._rank_codomainis a pair of reciprocal rank and unrank functions between the codomain andrange(parent._n).
- classmethod from_dict(t, parent, d)[source]¶
Create a
FiniteSetMapfrom a dictionary.Warning
no check is performed !
- classmethod from_list(t, parent, lst)[source]¶
Create a
FiniteSetMapfrom a list.Warning
no check is performed !
- getimage(i)[source]¶
Return the image of
ibyself.INPUT:
i– integer
EXAMPLES:
sage: F = FiniteSetMaps(["a", "b", "c", "d"], ["u", "v", "w"]) sage: fs = F._from_list_([1, 0, 2, 1]) sage: list(map(fs.getimage, ["a", "b", "c", "d"])) ['v', 'u', 'w', 'v']
>>> from sage.all import * >>> F = FiniteSetMaps(["a", "b", "c", "d"], ["u", "v", "w"]) >>> fs = F._from_list_([Integer(1), Integer(0), Integer(2), Integer(1)]) >>> list(map(fs.getimage, ["a", "b", "c", "d"])) ['v', 'u', 'w', 'v']
- image_set()[source]¶
Return the image set of
self.EXAMPLES:
sage: F = FiniteSetMaps(["a", "b", "c"]) sage: sorted(F.from_dict({"a": "b", "b": "a", "c": "b"}).image_set()) ['a', 'b'] sage: F = FiniteSetMaps(["a", "b", "c"]) sage: F(lambda x: "c").image_set() {'c'}
>>> from sage.all import * >>> F = FiniteSetMaps(["a", "b", "c"]) >>> sorted(F.from_dict({"a": "b", "b": "a", "c": "b"}).image_set()) ['a', 'b'] >>> F = FiniteSetMaps(["a", "b", "c"]) >>> F(lambda x: "c").image_set() {'c'}
- items()[source]¶
The items of
self.Return the list of the couple
(x, self(x))EXAMPLES:
sage: F = FiniteSetMaps(["a", "b", "c"]) sage: F.from_dict({"a": "b", "b": "a", "c": "b"}).items() [('a', 'b'), ('b', 'a'), ('c', 'b')]
>>> from sage.all import * >>> F = FiniteSetMaps(["a", "b", "c"]) >>> F.from_dict({"a": "b", "b": "a", "c": "b"}).items() [('a', 'b'), ('b', 'a'), ('c', 'b')]
- setimage(i, j)[source]¶
Set the image of
iasjinself.Warning
selfmust be mutable otherwise an exception is raised.INPUT:
i,j– twoobject’s
OUTPUT: none
EXAMPLES:
sage: F = FiniteSetMaps(["a", "b", "c", "d"], ["u", "v", "w"]) sage: fs = F(lambda x: "v") sage: fs2 = copy(fs) sage: fs2.setimage("a", "w") sage: fs2 map: a -> w, b -> v, c -> v, d -> v sage: with fs.clone() as fs3: ....: fs3.setimage("a", "u") ....: fs3.setimage("c", "w") sage: fs3 map: a -> u, b -> v, c -> w, d -> v
>>> from sage.all import * >>> F = FiniteSetMaps(["a", "b", "c", "d"], ["u", "v", "w"]) >>> fs = F(lambda x: "v") >>> fs2 = copy(fs) >>> fs2.setimage("a", "w") >>> fs2 map: a -> w, b -> v, c -> v, d -> v >>> with fs.clone() as fs3: ... fs3.setimage("a", "u") ... fs3.setimage("c", "w") >>> fs3 map: a -> u, b -> v, c -> w, d -> v
- sage.sets.finite_set_map_cy.FiniteSetMap_Set_from_dict(t, parent, d)[source]¶
Create a
FiniteSetMapfrom a dictionary.Warning
no check is performed !
- sage.sets.finite_set_map_cy.FiniteSetMap_Set_from_list(t, parent, lst)[source]¶
Create a
FiniteSetMapfrom a list.Warning
no check is performed !
- sage.sets.finite_set_map_cy.fibers(f, domain)[source]¶
Return the fibers of the function
fon the finite setdomain.INPUT:
f– a function or callabledomain– a finite iterable
OUTPUT:
a dictionary
dsuch thatd[y]is the set of allxindomainsuch thatf(x) = y
EXAMPLES:
sage: from sage.sets.finite_set_map_cy import fibers, fibers_args sage: fibers(lambda x: 1, []) {} sage: fibers(lambda x: x^2, [-1, 2, -3, 1, 3, 4]) {1: {1, -1}, 4: {2}, 9: {3, -3}, 16: {4}} sage: fibers(lambda x: 1, [-1, 2, -3, 1, 3, 4]) {1: {1, 2, 3, 4, -3, -1}} sage: fibers(lambda x: 1, [1,1,1]) {1: {1}}
>>> from sage.all import * >>> from sage.sets.finite_set_map_cy import fibers, fibers_args >>> fibers(lambda x: Integer(1), []) {} >>> fibers(lambda x: x**Integer(2), [-Integer(1), Integer(2), -Integer(3), Integer(1), Integer(3), Integer(4)]) {1: {1, -1}, 4: {2}, 9: {3, -3}, 16: {4}} >>> fibers(lambda x: Integer(1), [-Integer(1), Integer(2), -Integer(3), Integer(1), Integer(3), Integer(4)]) {1: {1, 2, 3, 4, -3, -1}} >>> fibers(lambda x: Integer(1), [Integer(1),Integer(1),Integer(1)]) {1: {1}}
See also
fibers_args()if one needs to pass extra arguments tof.
- sage.sets.finite_set_map_cy.fibers_args(f, domain, *args, **opts)[source]¶
Return the fibers of the function
fon the finite setdomain.It is the same as
fibers()except that one can pass extra argument forf(with a small overhead)EXAMPLES:
sage: from sage.sets.finite_set_map_cy import fibers_args sage: fibers_args(operator.pow, [-1, 2, -3, 1, 3, 4], 2) {1: {1, -1}, 4: {2}, 9: {3, -3}, 16: {4}}
>>> from sage.all import * >>> from sage.sets.finite_set_map_cy import fibers_args >>> fibers_args(operator.pow, [-Integer(1), Integer(2), -Integer(3), Integer(1), Integer(3), Integer(4)], Integer(2)) {1: {1, -1}, 4: {2}, 9: {3, -3}, 16: {4}}