# Multidimensional enumeration¶

AUTHORS:

• Joel B. Mohler (2006-10-12)
• William Stein (2006-07-19)
• Jon Hanke
sage.misc.mrange.cartesian_product_iterator(X)

Iterate over the Cartesian product.

INPUT:

• X - list or tuple of lists

OUTPUT: iterator over the cartesian product of the elements of X

EXAMPLES:

sage: list(cartesian_product_iterator([[1,2], ['a','b']]))
[(1, 'a'), (1, 'b'), (2, 'a'), (2, 'b')]
sage: list(cartesian_product_iterator([]))
[()]

sage.misc.mrange.mrange(sizes, typ=<type 'list'>)

Return the multirange list with given sizes and type.

This is the list version of xmrange. Use xmrange for the iterator.

More precisely, return the iterator over all objects of type typ of n-tuples of Python ints with entries between 0 and the integers in the sizes list. The iterator is empty if sizes is empty or contains any non-positive integer.

INPUT:

• sizes - a list of nonnegative integers
• typ - (default: list) a type or class; more generally, something that can be called with a list as input.

OUTPUT: a list

EXAMPLES:

sage: mrange([3,2])
[[0, 0], [0, 1], [1, 0], [1, 1], [2, 0], [2, 1]]
sage: mrange([3,2], tuple)
[(0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1)]
sage: mrange([3,2], sum)
[0, 1, 1, 2, 2, 3]


Examples that illustrate empty multi-ranges:

sage: mrange([5,3,-2])
[]
sage: mrange([5,3,0])
[]


This example isn’t empty, and shouldn’t be. See trac #6561.

sage: mrange([])
[[]]


AUTHORS:

• Jon Hanke
• William Stein
sage.misc.mrange.mrange_iter(iter_list, typ=<type 'list'>)

Return the multirange list derived from the given list of iterators.

This is the list version of xmrange_iter. Use xmrange_iter for the iterator.

More precisely, return the iterator over all objects of type typ of n-tuples of Python ints with entries between 0 and the integers in the sizes list. The iterator is empty if sizes is empty or contains any non-positive integer.

INPUT:

• iter_list - a finite iterable of finite iterables
• typ - (default: list) a type or class; more generally, something that can be called with a list as input.

OUTPUT: a list

EXAMPLES:

sage: mrange_iter([range(3),[0,2]])
[[0, 0], [0, 2], [1, 0], [1, 2], [2, 0], [2, 2]]
sage: mrange_iter([['Monty','Flying'],['Python','Circus']], tuple)
[('Monty', 'Python'), ('Monty', 'Circus'), ('Flying', 'Python'), ('Flying', 'Circus')]
sage: mrange_iter([[2,3,5,7],[1,2]], sum)
[3, 4, 4, 5, 6, 7, 8, 9]


Examples that illustrate empty multi-ranges:

sage: mrange_iter([range(5),xrange(3),xrange(-2)])
[]
sage: mrange_iter([range(5),range(3),range(0)])
[]


This example isn’t empty, and shouldn’t be. See trac #6561.

sage: mrange_iter([])
[[]]


AUTHORS:

• Joel B. Mohler
class sage.misc.mrange.xmrange(sizes, typ=<type 'list'>)

Return the multirange iterate with given sizes and type.

More precisely, return the iterator over all objects of type typ of n-tuples of Python ints with entries between 0 and the integers in the sizes list. The iterator is empty if sizes is empty or contains any non-positive integer.

Use mrange for the non-iterator form.

INPUT:

• sizes - a list of nonnegative integers
• typ - (default: list) a type or class; more generally, something that can be called with a list as input.

OUTPUT: a generator

EXAMPLES: We create multi-range iterators, print them and also iterate through a tuple version.

sage: z = xmrange([3,2]);z
xmrange([3, 2])
sage: z = xmrange([3,2], tuple);z
xmrange([3, 2], <type 'tuple'>)
sage: for a in z:
...    print a
(0, 0)
(0, 1)
(1, 0)
(1, 1)
(2, 0)
(2, 1)


We illustrate a few more iterations.

sage: list(xmrange([3,2]))
[[0, 0], [0, 1], [1, 0], [1, 1], [2, 0], [2, 1]]
sage: list(xmrange([3,2], tuple))
[(0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1)]


Here we compute the sum of each element of the multi-range iterator:

sage: list(xmrange([3,2], sum))
[0, 1, 1, 2, 2, 3]


Next we compute the product:

sage: list(xmrange([3,2], prod))
[0, 0, 0, 1, 0, 2]


Examples that illustrate empty multi-ranges.

sage: list(xmrange([5,3,-2]))
[]
sage: list(xmrange([5,3,0]))
[]


This example isn’t empty, and shouldn’t be. See trac #6561.

sage: list(xmrange([]))
[[]]


We use a multi-range iterator to iterate through the Cartesian product of sets.

sage: X = ['red', 'apple', 389]
sage: Y = ['orange', 'horse']
sage: for i,j in xmrange([len(X), len(Y)]):
...    print (X[i], Y[j])
('red', 'orange')
('red', 'horse')
('apple', 'orange')
('apple', 'horse')
(389, 'orange')
(389, 'horse')


AUTHORS:

• Jon Hanke
• William Stein
class sage.misc.mrange.xmrange_iter(iter_list, typ=<type 'list'>)

Return the multirange iterate derived from the given iterators and type.

Note

This basically gives you the Cartesian product of sets.

More precisely, return the iterator over all objects of type typ of n-tuples of Python ints with entries between 0 and the integers in the sizes list. The iterator is empty if sizes is empty or contains any non-positive integer.

Use mrange_iter for the non-iterator form.

INPUT:

• iter_list - a list of objects usable as iterators (possibly

lists)

• typ - (default: list) a type or class; more generally,

something that can be called with a list as input.

OUTPUT: a generator

EXAMPLES: We create multi-range iterators, print them and also iterate through a tuple version.

sage: z = xmrange_iter([xrange(3),xrange(2)]);z
xmrange_iter([xrange(3), xrange(2)])
sage: z = xmrange_iter([range(3),range(2)], tuple);z
xmrange_iter([[0, 1, 2], [0, 1]], <type 'tuple'>)
sage: for a in z:
...    print a
(0, 0)
(0, 1)
(1, 0)
(1, 1)
(2, 0)
(2, 1)


We illustrate a few more iterations.

sage: list(xmrange_iter([range(3),range(2)]))
[[0, 0], [0, 1], [1, 0], [1, 1], [2, 0], [2, 1]]
sage: list(xmrange_iter([range(3),range(2)], tuple))
[(0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1)]


Here we compute the sum of each element of the multi-range iterator:

sage: list(xmrange_iter([range(3),range(2)], sum))
[0, 1, 1, 2, 2, 3]


Next we compute the product:

sage: list(xmrange_iter([range(3),range(2)], prod))
[0, 0, 0, 1, 0, 2]


Examples that illustrate empty multi-ranges.

sage: list(xmrange_iter([xrange(5),xrange(3),xrange(-2)]))
[]
sage: list(xmrange_iter([xrange(5),xrange(3),xrange(0)]))
[]


This example isn’t empty, and shouldn’t be. See trac #6561.

sage: list(xmrange_iter([]))
[[]]


We use a multi-range iterator to iterate through the Cartesian product of sets.

sage: X = ['red', 'apple', 389]
sage: Y = ['orange', 'horse']
sage: for i,j in xmrange_iter([X, Y], tuple):
...    print (i, j)
('red', 'orange')
('red', 'horse')
('apple', 'orange')
('apple', 'horse')
(389, 'orange')
(389, 'horse')


AUTHORS:

• Joel B. Mohler
cardinality()

Return the cardinality of this iterator.

EXAMPLES:

sage: C = cartesian_product_iterator([xrange(3), xrange(4)])
sage: C.cardinality()
12
sage: C = cartesian_product_iterator([ZZ,QQ])
sage: C.cardinality()
+Infinity
sage: C = cartesian_product_iterator([ZZ,[]])
sage: C.cardinality()
0


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