Backtracking¶
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.
GenericBacktracker: Depth first search through a tree described by achildrenfunction, with branch pruning, etc.
Deprecated classes (use RecursivelyEnumeratedSet() instead):
SearchForest: Depth and breadth first search through a tree described by achildrenfunction.TransitiveIdeal: Depth first search through a graph described by aneighboursrelation.TransitiveIdealGraded: Breadth first search through a graph described by aneighboursrelation.
Deprecation details:
SearchForest(seeds, succ)keeps the same behavior as before trac ticket #6637 and is now the same asRecursivelyEnumeratedSet(seeds, succ, structure='forest', enumeration='depth').TransitiveIdeal(succ, seeds)keeps the same behavior as before trac ticket #6637 and is now the same asRecursivelyEnumeratedSet(seeds, succ, structure=None, enumeration='naive').TransitiveIdealGraded(succ, seeds, max_depth)keeps the same behavior as before trac ticket #6637 and is now the same asRecursivelyEnumeratedSet(seeds, succ, structure=None, enumeration='breadth', max_depth=max_depth).
Todo
- For now the code of
SearchForestis still insage/combinat/backtrack.py. It should be moved insage/sets/recursively_enumerated_set.pyxinto a class namedRecursivelyEnumeratedSet_forestin a later ticket. - Deprecate
TransitiveIdealandTransitiveIdealGraded. - Once the deprecation has been there for enough time: delete
TransitiveIdealandTransitiveIdealGraded.
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class
sage.combinat.backtrack.GenericBacktracker(initial_data, initial_state)¶ Bases:
objectA generic backtrack tool for exploring a search space organized as a tree, with branch pruning, etc.
See also
SearchForestandTransitiveIdealfor handling simple special cases.
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class
sage.combinat.backtrack.PositiveIntegerSemigroup¶ Bases:
sage.structure.unique_representation.UniqueRepresentation,sage.combinat.backtrack.SearchForestThe commutative additive semigroup of positive integers.
This class provides an example of algebraic structure which inherits from
SearchForest. It builds the positive integers a la Peano, and endows it with its natural commutative additive semigroup structure.EXAMPLES:
sage: from sage.combinat.backtrack import PositiveIntegerSemigroup sage: PP = PositiveIntegerSemigroup() sage: PP.category() Join of Category of monoids and Category of commutative additive semigroups and Category of infinite enumerated sets and Category of facade sets sage: PP.cardinality() +Infinity sage: PP.one() 1 sage: PP.an_element() 1 sage: some_elements = list(PP.some_elements()); some_elements [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100]
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children(x)¶ Return the single child
x+1of the integerxEXAMPLES:
sage: from sage.combinat.backtrack import PositiveIntegerSemigroup sage: PP = PositiveIntegerSemigroup() sage: list(PP.children(1)) [2] sage: list(PP.children(42)) [43]
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one()¶ Return the unit of
self.EXAMPLES:
sage: from sage.combinat.backtrack import PositiveIntegerSemigroup sage: PP = PositiveIntegerSemigroup() sage: PP.one() 1
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roots()¶ Return the single root of
self.EXAMPLES:
sage: from sage.combinat.backtrack import PositiveIntegerSemigroup sage: PP = PositiveIntegerSemigroup() sage: list(PP.roots()) [1]
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class
sage.combinat.backtrack.SearchForest(roots=None, children=None, post_process=None, algorithm='depth', facade=None, category=None)¶ Bases:
sage.structure.parent.ParentThe enumerated set of the nodes of the forest having the given
roots, and wherechildren(x)returns the children of the nodexof the forest.See also
GenericBacktracker,TransitiveIdeal, andTransitiveIdealGraded.INPUT:
roots– a list (or iterable)children– a function returning a list (or iterable, or iterator)post_process– a function defined over the nodes of the forest (default: no post processing)algorithm–'depth'or'breadth'(default:'depth')category– a category (default:EnumeratedSets)
The option
post_processallows for customizing the nodes that are actually produced. Furthermore, iff(x)returnsNone, thenxwon’t be output at all.EXAMPLES:
We construct the set of all binary sequences of length at most three, and list them:
sage: from sage.combinat.backtrack import SearchForest sage: S = SearchForest( [[]], ....: lambda l: [l+[0], l+[1]] if len(l) < 3 else [], ....: category=FiniteEnumeratedSets()) sage: S.list() [[], [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]]
SearchForestneeds to be explicitly told that the set is finite for the following to work:sage: S.category() Category of finite enumerated sets sage: S.cardinality() 15
We proceed with the set of all lists of letters in
0,1,2without repetitions, ordered by increasing length (i.e. using a breadth first search through the tree):sage: from sage.combinat.backtrack import SearchForest sage: tb = SearchForest( [[]], ....: lambda l: [l + [i] for i in range(3) if i not in l], ....: algorithm = 'breadth', ....: category=FiniteEnumeratedSets()) sage: tb[0] [] sage: tb.cardinality() 16 sage: list(tb) [[], [0], [1], [2], [0, 1], [0, 2], [1, 0], [1, 2], [2, 0], [2, 1], [0, 1, 2], [0, 2, 1], [1, 0, 2], [1, 2, 0], [2, 0, 1], [2, 1, 0]]
For infinite sets, this option should be set carefully to ensure that all elements are actually generated. The following example builds the set of all ordered pairs \((i,j)\) of nonnegative integers such that \(j\leq 1\):
sage: from sage.combinat.backtrack import SearchForest sage: I = SearchForest([(0,0)], ....: lambda l: [(l[0]+1, l[1]), (l[0], 1)] ....: if l[1] == 0 else [(l[0], l[1]+1)])
With a depth first search, only the elements of the form \((i,0)\) are generated:
sage: depth_search = I.depth_first_search_iterator() sage: [next(depth_search) for i in range(7)] [(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (5, 0), (6, 0)]
Using instead breadth first search gives the usual anti-diagonal iterator:
sage: breadth_search = I.breadth_first_search_iterator() sage: [next(breadth_search) for i in range(15)] [(0, 0), (1, 0), (0, 1), (2, 0), (1, 1), (0, 2), (3, 0), (2, 1), (1, 2), (0, 3), (4, 0), (3, 1), (2, 2), (1, 3), (0, 4)]
Deriving subclasses
The class of a parent \(A\) may derive from
SearchForestso that \(A\) can benefit from enumeration tools. As a running example, we consider the problem of enumerating integers whose binary expansion have at most three nonzero digits. For example, \(3 = 2^1 + 2^0\) has two nonzero digits. \(15 = 2^3 + 2^2 + 2^1 + 2^0\) has four nonzero digits. In fact, \(15\) is the smallest integer which is not in the enumerated set.To achieve this, we use
SearchForestto enumerate binary tuples with at most three nonzero digits, apply a post processing to recover the corresponding integers, and discard tuples finishing by zero.A first approach is to pass the
rootsandchildrenfunctions as arguments toSearchForest.__init__():sage: from sage.combinat.backtrack import SearchForest sage: class A(UniqueRepresentation, SearchForest): ....: def __init__(self): ....: SearchForest.__init__(self, [()], ....: lambda x : [x+(0,), x+(1,)] if sum(x) < 3 else [], ....: lambda x : sum(x[i]*2^i for i in range(len(x))) if sum(x) != 0 and x[-1] != 0 else None, ....: algorithm = 'breadth', ....: category=InfiniteEnumeratedSets()) sage: MyForest = A(); MyForest An enumerated set with a forest structure sage: MyForest.category() Category of infinite enumerated sets sage: p = iter(MyForest) sage: [next(p) for i in range(30)] [1, 2, 3, 4, 6, 5, 7, 8, 12, 10, 14, 9, 13, 11, 16, 24, 20, 28, 18, 26, 22, 17, 25, 21, 19, 32, 48, 40, 56, 36]
An alternative approach is to implement
rootsandchildrenas methods of the subclass (in fact they could also be attributes of \(A\)). Namely,A.roots()must return an iterable containing the enumeration generators, andA.children(x)must return an iterable over the children of \(x\). Optionally, \(A\) can have a method or attribute such thatA.post_process(x)returns the desired output for the nodexof the tree:sage: from sage.combinat.backtrack import SearchForest sage: class A(UniqueRepresentation, SearchForest): ....: def __init__(self): ....: SearchForest.__init__(self, algorithm = 'breadth', ....: category=InfiniteEnumeratedSets()) ....: ....: def roots(self): ....: return [()] ....: ....: def children(self, x): ....: if sum(x) < 3: ....: return [x+(0,), x+(1,)] ....: else: ....: return [] ....: ....: def post_process(self, x): ....: if sum(x) == 0 or x[-1] == 0: ....: return None ....: else: ....: return sum(x[i]*2^i for i in range(len(x))) sage: MyForest = A(); MyForest An enumerated set with a forest structure sage: MyForest.category() Category of infinite enumerated sets sage: p = iter(MyForest) sage: [next(p) for i in range(30)] [1, 2, 3, 4, 6, 5, 7, 8, 12, 10, 14, 9, 13, 11, 16, 24, 20, 28, 18, 26, 22, 17, 25, 21, 19, 32, 48, 40, 56, 36]
Warning
A
SearchForestinstance is picklable if and only if the input functions are themselves picklable. This excludes anonymous or interactively defined functions:sage: def children(x): ....: return [x+1] sage: S = SearchForest( [1], children, category=InfiniteEnumeratedSets()) sage: dumps(S) Traceback (most recent call last): ... PicklingError: Can't pickle <...function...>: attribute lookup ... failed
Let us now fake
childrenbeing defined in a Python module:sage: import __main__ sage: __main__.children = children sage: S = SearchForest( [1], children, category=InfiniteEnumeratedSets()) sage: loads(dumps(S)) An enumerated set with a forest structure
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breadth_first_search_iterator()¶ Return a breadth first search iterator over the elements of
selfEXAMPLES:
sage: from sage.combinat.backtrack import SearchForest sage: f = SearchForest([[]], ....: lambda l: [l+[0], l+[1]] if len(l) < 3 else []) sage: list(f.breadth_first_search_iterator()) [[], [0], [1], [0, 0], [0, 1], [1, 0], [1, 1], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, 0, 0], [1, 0, 1], [1, 1, 0], [1, 1, 1]] sage: S = SearchForest([(0,0)], ....: lambda x : [(x[0], x[1]+1)] if x[1] != 0 else [(x[0]+1,0), (x[0],1)], ....: post_process = lambda x: x if ((is_prime(x[0]) and is_prime(x[1])) and ((x[0] - x[1]) == 2)) else None) sage: p = S.breadth_first_search_iterator() sage: [next(p), next(p), next(p), next(p), next(p), next(p), next(p)] [(5, 3), (7, 5), (13, 11), (19, 17), (31, 29), (43, 41), (61, 59)]
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children(x)¶ Return the children of the element
xThe result can be a list, an iterable, an iterator, or even a generator.
EXAMPLES:
sage: from sage.combinat.backtrack import SearchForest sage: I = SearchForest([(0,0)], lambda l: [(l[0]+1, l[1]), (l[0], 1)] if l[1] == 0 else [(l[0], l[1]+1)]) sage: [i for i in I.children((0,0))] [(1, 0), (0, 1)] sage: [i for i in I.children((1,0))] [(2, 0), (1, 1)] sage: [i for i in I.children((1,1))] [(1, 2)] sage: [i for i in I.children((4,1))] [(4, 2)] sage: [i for i in I.children((4,0))] [(5, 0), (4, 1)]
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depth_first_search_iterator()¶ Return a depth first search iterator over the elements of
selfEXAMPLES:
sage: from sage.combinat.backtrack import SearchForest sage: f = SearchForest([[]], ....: lambda l: [l+[0], l+[1]] if len(l) < 3 else []) sage: list(f.depth_first_search_iterator()) [[], [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]]
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elements_of_depth_iterator(depth=0)¶ Return an iterator over the elements of
selfof given depth. An element of depth \(n\) can be obtained applying \(n\) times the children function from a root.EXAMPLES:
sage: from sage.combinat.backtrack import SearchForest sage: S = SearchForest([(0,0)] , ....: lambda x : [(x[0], x[1]+1)] if x[1] != 0 else [(x[0]+1,0), (x[0],1)], ....: post_process = lambda x: x if ((is_prime(x[0]) and is_prime(x[1])) ....: and ((x[0] - x[1]) == 2)) else None) sage: p = S.elements_of_depth_iterator(8) sage: next(p) (5, 3) sage: S = SearchForest(NN, lambda x : [], ....: lambda x: x^2 if x.is_prime() else None) sage: p = S.elements_of_depth_iterator(0) sage: [next(p), next(p), next(p), next(p), next(p)] [4, 9, 25, 49, 121]
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map_reduce(map_function=None, reduce_function=None, reduce_init=None)¶ Apply a Map/Reduce algorithm on
selfINPUT:
map_function– a function from the element ofselfto some set with a reduce operation (e.g.: a monoid). The default value is the constant function1.reduce_function– the reduce function (e.g.: the addition of a monoid). The default value is+.reduce_init– the initialisation of the reduction (e.g.: the neutral element of the monoid). The default value is0.
Note
the effect of the default values is to compute the cardinality of
self.EXAMPLES:
sage: seeds = [([i],i, i) for i in range(1,10)] sage: def succ(t): ....: list, sum, last = t ....: return [(list + [i], sum + i, i) for i in range(1, last)] sage: F = RecursivelyEnumeratedSet(seeds, succ, ....: structure='forest', enumeration='depth') sage: y = var('y') sage: def map_function(t): ....: li, sum, _ = t ....: return y ^ sum sage: reduce_function = lambda x,y: x + y sage: F.map_reduce(map_function, reduce_function, 0) y^45 + y^44 + y^43 + 2*y^42 + 2*y^41 + 3*y^40 + 4*y^39 + 5*y^38 + 6*y^37 + 8*y^36 + 9*y^35 + 10*y^34 + 12*y^33 + 13*y^32 + 15*y^31 + 17*y^30 + 18*y^29 + 19*y^28 + 21*y^27 + 21*y^26 + 22*y^25 + 23*y^24 + 23*y^23 + 23*y^22 + 23*y^21 + 22*y^20 + 21*y^19 + 21*y^18 + 19*y^17 + 18*y^16 + 17*y^15 + 15*y^14 + 13*y^13 + 12*y^12 + 10*y^11 + 9*y^10 + 8*y^9 + 6*y^8 + 5*y^7 + 4*y^6 + 3*y^5 + 2*y^4 + 2*y^3 + y^2 + y
Here is an example with the default values:
sage: F.map_reduce() 511
See also
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roots()¶ Return an iterable over the roots of
self.EXAMPLES:
sage: from sage.combinat.backtrack import SearchForest sage: I = SearchForest([(0,0)], lambda l: [(l[0]+1, l[1]), (l[0], 1)] if l[1] == 0 else [(l[0], l[1]+1)]) sage: [i for i in I.roots()] [(0, 0)] sage: I = SearchForest([(0,0),(1,1)], lambda l: [(l[0]+1, l[1]), (l[0], 1)] if l[1] == 0 else [(l[0], l[1]+1)]) sage: [i for i in I.roots()] [(0, 0), (1, 1)]
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class
sage.combinat.backtrack.TransitiveIdeal(succ, generators)¶ Bases:
sage.sets.recursively_enumerated_set.RecursivelyEnumeratedSet_genericGeneric tool for constructing ideals of a relation.
INPUT:
relation– a function (or callable) returning a list (or iterable)generators– a list (or iterable)
Returns the set \(S\) of elements that can be obtained by repeated application of
relationon the elements ofgenerators.Consider
relationas modeling a directed graph (possibly with loops, cycles, or circuits). Then \(S\) is the ideal generated bygeneratorsunder this relation.Enumerating the elements of \(S\) is achieved by depth first search through the graph. The time complexity is \(O(n+m)\) where \(n\) is the size of the ideal, and \(m\) 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 \(S\).
See also
SearchForestandTransitiveIdealGraded.EXAMPLES:
sage: from sage.combinat.backtrack import TransitiveIdeal 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: sorted(i for i in TransitiveIdeal(lambda i: [mod(i+3,10),mod(i+5,10)], [0])) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] sage: sorted(i for i in TransitiveIdeal(lambda i: [mod(i+4,10),mod(i+6,10)], [0])) [0, 2, 4, 6, 8] 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: [ next(f) for i in range(6) ] [0, 1, 2, 3, 4, 5]
We compute all the permutations of 3:
sage: sorted(p for p in TransitiveIdeal(attrcall("permutohedron_succ"), [Permutation([1,2,3])])) [[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]]
We compute all the permutations which are larger than [3,1,2,4], [2,1,3,4] in the right permutohedron:
sage: sorted(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, 3, 1, 4], [2, 3, 4, 1], [2, 4, 1, 3], [2, 4, 3, 1], [3, 1, 2, 4], [3, 1, 4, 2], [3, 2, 1, 4], [3, 2, 4, 1], [3, 4, 1, 2], [3, 4, 2, 1], [4, 2, 1, 3], [4, 2, 3, 1], [4, 3, 1, 2], [4, 3, 2, 1]]
Using TransitiveIdeal people have been using the
__contains__method provided from the__iter__method. We need to make sure that this continues to work:sage: T = TransitiveIdeal(lambda a:[a+7,a+5], [0]) sage: 12 in T True
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class
sage.combinat.backtrack.TransitiveIdealGraded(succ, generators, max_depth=inf)¶ Bases:
sage.sets.recursively_enumerated_set.RecursivelyEnumeratedSet_genericGeneric tool for constructing ideals of a relation.
INPUT:
relation– a function (or callable) returning a list (or iterable)generators– a list (or iterable)max_depth– (Default: infinity) Specifies the maximal depth to which elements are computed
Return the set \(S\) of elements that can be obtained by repeated application of
relationon the elements ofgenerators.Consider
relationas modeling a directed graph (possibly with loops, cycles, or circuits). Then \(S\) is the ideal generated bygeneratorsunder this relation.Enumerating the elements of \(S\) is achieved by breadth first search through the graph; hence elements are enumerated by increasing distance from the generators. The time complexity is \(O(n+m)\) where \(n\) is the size of the ideal, and \(m\) 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 \(S\).
See also
SearchForestandTransitiveIdeal.EXAMPLES:
sage: from sage.combinat.backtrack import TransitiveIdealGraded 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 illustrate that the enumeration is done lazily, by breadth 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([ next(f) for i in range(3) ]) [-10, 0, 10]
The elements at distance 1 from the generators:
sage: sorted([ next(f) for i in range(6) ]) [-11, -9, -1, 1, 9, 11]
The elements at distance 2 from the generators:
sage: sorted([ next(f) 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: sorted(p for p in TransitiveIdealGraded(attrcall("permutohedron_succ"), ....: [Permutation([3,1,2,4]), Permutation([2,1,3,4])])) [[2, 1, 3, 4], [2, 1, 4, 3], [2, 3, 1, 4], [2, 3, 4, 1], [2, 4, 1, 3], [2, 4, 3, 1], [3, 1, 2, 4], [3, 1, 4, 2], [3, 2, 1, 4], [3, 2, 4, 1], [3, 4, 1, 2], [3, 4, 2, 1], [4, 2, 1, 3], [4, 2, 3, 1], [4, 3, 1, 2], [4, 3, 2, 1]]
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sage.combinat.backtrack.search_forest_iterator(roots, children, algorithm='depth')¶ Return an iterator on the nodes of the forest having the given roots, and where
children(x)returns the children of the nodexof the forest. Note that every node of the tree is returned, not simply the leaves.INPUT:
roots– a list (or iterable)children– a function returning a list (or iterable)algorithm–'depth'or'breadth'(default:'depth')
EXAMPLES:
We construct the prefix tree of binary sequences of length at most three, and enumerate its nodes:
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]]
By default, the nodes are iterated through by depth first search. We can instead use a breadth first search (increasing depth):
sage: list(search_forest_iterator([[]], lambda l: [l+[0], l+[1]] ....: if len(l) < 3 else [], ....: algorithm='breadth')) [[], [0], [1], [0, 0], [0, 1], [1, 0], [1, 1], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, 0, 0], [1, 0, 1], [1, 1, 0], [1, 1, 1]]
This allows for iterating trough trees of infinite depth:
sage: it = search_forest_iterator([[]], lambda l: [l+[0], l+[1]], algorithm='breadth') sage: [ next(it) for i in range(16) ] [[], [0], [1], [0, 0], [0, 1], [1, 0], [1, 1], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, 0, 0], [1, 0, 1], [1, 1, 0], [1, 1, 1], [0, 0, 0, 0]]
Here is an iterator through the prefix tree of sequences of letters in \(0,1,2\) without repetitions, sorted by length; the leaves are therefore permutations:
sage: list(search_forest_iterator([[]], lambda l: [l + [i] for i in range(3) if i not in l], ....: algorithm='breadth')) [[], [0], [1], [2], [0, 1], [0, 2], [1, 0], [1, 2], [2, 0], [2, 1], [0, 1, 2], [0, 2, 1], [1, 0, 2], [1, 2, 0], [2, 0, 1], [2, 1, 0]]