This library contains generic tools for constructing large sets whose elements can be enumerated by exploring a search space with a (lazy) tree or graph structure.
TODO:
A generic backtrack tool for exploring a search space organized as a tree, with branch pruning, etc.
See also SearchForest and TransitiveIdeal for handling simple special cases.
EXAMPLES:
sage: from sage.combinat.backtrack import GenericBacktracker
sage: p = GenericBacktracker([], 1)
sage: loads(dumps(p))
<sage.combinat.backtrack.GenericBacktracker object at 0x...>
EXAMPLES:
sage: from sage.combinat.permutation import PatternAvoider
sage: p = PatternAvoider(4, [[1,3,2]])
sage: len(list(p))
14
Returns the set of nodes of the forest having the given roots, and where children(x) returns the children of the node x of the forest.
See also GenericBacktracker, TransitiveIdeal, and TransitiveIdealGraded.
INPUT:
EXAMPLES:
A generator object for binary sequences of length 3, listed:
sage: list(SearchForest([[]], lambda l: [l+[0], l+[1]] if len(l) < 3 else []))
[[], [0], [0, 0], [0, 0, 0], [0, 0, 1], [0, 1], [0, 1, 0], [0, 1, 1], [1], [1, 0], [1, 0, 0], [1, 0, 1], [1, 1], [1, 1, 0], [1, 1, 1]]
A generator object for ordered sequences of length 2 from a 4-set, sampled:
sage: tb = SearchForest([[]], lambda l: [l + [i] for i in range(4) if i not in l] if len(l) < 2 else [])
sage: tb[0]
[]
sage: tb[16]
[3, 2]
TESTS:
sage: C = SearchForest((1,), lambda x: [x+1])
sage: C._roots
(1,)
sage: C._children
<function <lambda> at ...>
Returns an iterator on the elements of self.
EXAMPLES:
sage: def succ(l):
... return [l+[0], l+[1]]
...
sage: C = SearchForest(([],), succ)
sage: f = C.__iter__()
sage: f.next()
[]
sage: f.next()
[0]
sage: f.next()
[0, 0]
sage: import __main__
sage: __main__.succ = succ # just because succ has been defined interactively
sage: loads(dumps(C))
An enumerated set
TESTS:
sage: SearchForest((1,), lambda x: [x+1]) # Todo: improve!
An enumerated set
Generic tool for constructing ideals of a relation.
INPUT:
Returns the set of elements that can be obtained by repeated
application of relation on the elements of generators.
Consider relation as modeling a directed graph (possibly with
loops, cycles, or circuits). Then is the ideal generated by
generators under this relation.
Enumerating the elements of is achieved by depth first search
through the graph. The time complexity is
where
is
the size of the ideal, and
the number of edges in the
relation. The memory complexity is the depth, that is the maximal
distance between a generator and an element of
.
See also SearchForest and TransitiveIdealGraded.
EXAMPLES:
sage: [i for i in TransitiveIdeal(lambda i: [i+1] if i<10 else [], [0])]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
sage: [i for i in TransitiveIdeal(lambda i: [mod(i+1,3)], [0])]
[0, 1, 2]
sage: [i for i in TransitiveIdeal(lambda i: [mod(i+2,3)], [0])]
[0, 2, 1]
sage: [i for i in TransitiveIdeal(lambda i: [mod(i+2,10)], [0])]
[0, 2, 4, 6, 8]
sage: [i for i in TransitiveIdeal(lambda i: [mod(i+3,10),mod(i+5,10)], [0])]
[0, 3, 8, 1, 4, 5, 6, 7, 9, 2]
sage: [i for i in TransitiveIdeal(lambda i: [mod(i+4,10),mod(i+6,10)], [0])]
[0, 4, 8, 2, 6]
sage: [i for i in TransitiveIdeal(lambda i: [mod(i+3,9)], [0,1])]
[0, 1, 3, 4, 6, 7]
sage: [p for p in TransitiveIdeal(lambda x:[x],[Permutation([3,1,2,4]), Permutation([2,1,3,4])])]
[[2, 1, 3, 4], [3, 1, 2, 4]]
We now illustrate that the enumeration is done lazily, by depth first search:
sage: C = TransitiveIdeal(lambda x: [x-1, x+1], (-10, 0, 10))
sage: f = C.__iter__()
sage: [ f.next() for i in range(6) ]
[0, 1, 2, 3, 4, 5]
We compute all the permutations of 3:
sage: [p for p in TransitiveIdeal(attrcall("permutohedron_succ"), [Permutation([1,2,3])])]
[[1, 2, 3], [2, 1, 3], [1, 3, 2], [2, 3, 1], [3, 2, 1], [3, 1, 2]]
We compute all the permutations which are larger than [3,1,2,4], [2,1,3,4] in the right permutohedron:
sage: [p for p in TransitiveIdeal(attrcall("permutohedron_succ"), [Permutation([3,1,2,4]), Permutation([2,1,3,4])])]
[[2, 1, 3, 4], [2, 1, 4, 3], [2, 4, 1, 3], [4, 2, 1, 3], [4, 2, 3, 1], [4, 3, 2, 1], [3, 1, 2, 4], [2, 4, 3, 1], [3, 2, 1, 4], [2, 3, 1, 4], [2, 3, 4, 1], [3, 2, 4, 1], [3, 1, 4, 2], [3, 4, 2, 1], [3, 4, 1, 2], [4, 3, 1, 2]]
TESTS:
sage: C = TransitiveIdeal(factor, (1, 2, 3))
sage: C._succ
<function factor at ...>
sage: C._generators
(1, 2, 3)
sage: loads(dumps(C)) # should test for equality with C, but equality is not implemented
Returns an iterator on the elements of self.
TESTS:
sage: C = TransitiveIdeal(lambda x: [1,2], ())
sage: list(C) # indirect doctest
[]
sage: C = TransitiveIdeal(lambda x: [1,2], (1,))
sage: list(C) # indirect doctest
[1, 2]
sage: C = TransitiveIdeal(lambda x: [], (1,2))
sage: list(C) # indirect doctest
[1, 2]
Generic tool for constructing ideals of a relation.
INPUT:
Returns the set of elements that can be obtained by repeated
application of relation on the elements of generators.
Consider relation as modeling a directed graph (possibly with
loops, cycles, or circuits). Then is the ideal generated by
generators under this relation.
Enumerating the elements of is achieved by breath first search
through the graph; hence elements are enumerated by increasing
distance from the generators. The time complexity is
where
is the size of the ideal, and
the number of edges in
the relation. The memory complexity is the depth, that is the
maximal distance between a generator and an element of
.
See also SearchForest and TransitiveIdeal.
EXAMPLES:
sage: [i for i in TransitiveIdealGraded(lambda i: [i+1] if i<10 else [], [0])]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
We now illustrates that the enumeration is done lazily, by breath first search:
sage: C = TransitiveIdealGraded(lambda x: [x-1, x+1], (-10, 0, 10))
sage: f = C.__iter__()
The elements at distance 0 from the generators:
sage: sorted([ f.next() for i in range(3) ])
[-10, 0, 10]
The elements at distance 1 from the generators:
sage: sorted([ f.next() for i in range(6) ])
[-11, -9, -1, 1, 9, 11]
The elements at distance 2 from the generators:
sage: sorted([ f.next() for i in range(6) ])
[-12, -8, -2, 2, 8, 12]
The enumeration order between elements at the same distance is not specified.
We compute all the permutations which are larger than [3,1,2,4] or [2,1,3,4] in the permutohedron:
sage: [p for p in TransitiveIdealGraded(attrcall("permutohedron_succ"), [Permutation([3,1,2,4]), Permutation([2,1,3,4])])]
[[3, 1, 2, 4], [2, 1, 3, 4], [2, 1, 4, 3], [3, 2, 1, 4], [2, 3, 1, 4], [3, 1, 4, 2], [2, 3, 4, 1], [3, 4, 1, 2], [3, 2, 4, 1], [2, 4, 1, 3], [2, 4, 3, 1], [4, 3, 1, 2], [4, 2, 1, 3], [3, 4, 2, 1], [4, 2, 3, 1], [4, 3, 2, 1]]
TESTS:
sage: C = TransitiveIdealGraded(factor, (1, 2, 3))
sage: C._succ
<function factor at ...>
sage: C._generators
(1, 2, 3)
sage: loads(dumps(C)) # should test for equality with C, but equality is not implemented
Returns an iterator on the elements of self.
TESTS:
sage: C = TransitiveIdeal(lambda x: [1,2], ())
sage: list(C) # indirect doctest
[]
sage: C = TransitiveIdeal(lambda x: [1,2], (1,))
sage: list(C) # indirect doctest
[1, 2]
sage: C = TransitiveIdeal(lambda x: [], (1,2))
sage: list(C) # indirect doctest
[1, 2]
Returns an iterator on the nodes of the forest having the given roots, and where children(x) returns the children of the node x of the forest. Note that every node of the tree is returned, not simply the leaves.
INPUT:
EXAMPLES:
Search tree where leaves are binary sequences of length 3:
sage: from sage.combinat.backtrack import search_forest_iterator
sage: list(search_forest_iterator([[]], lambda l: [l+[0], l+[1]] if len(l) < 3 else []))
[[], [0], [0, 0], [0, 0, 0], [0, 0, 1], [0, 1], [0, 1, 0], [0, 1, 1], [1], [1, 0], [1, 0, 0], [1, 0, 1], [1, 1], [1, 1, 0], [1, 1, 1]]
Search tree where leaves are ordered sequences of length 2 from a 4-set:
sage: from sage.combinat.backtrack import search_forest_iterator
sage: list(search_forest_iterator([[]], lambda l: [l + [i] for i in range(4) if i not in l] if len(l) < 2 else []))
[[], [0], [0, 1], [0, 2], [0, 3], [1], [1, 0], [1, 2], [1, 3], [2], [2, 0], [2, 1], [2, 3], [3], [3, 0], [3, 1], [3, 2]]