# Gray codes¶

REFERENCES:

 [Knuth-TAOCP2A] D. Knuth “The art of computer programming”, fascicules 2A, “generating all n-tuples”
 [Knuth-TAOCP3A] D. Knuth “The art of computer programming”, fascicule 3A “generating all combinations”

## Functions¶

sage.combinat.gray_codes.combinations(n, t)

Iterator through the switches of the revolving door algorithm.

The revolving door algorithm is a way to generate all combinations of a set (i.e. the subset of given cardinality) in such way that two consecutive subsets differ by one element. At each step, the iterator output a pair (i,j) where the item i has to be removed and j has to be added.

The ground set is always $$\{0, 1, ..., n-1\}$$. Note that n can be infinity in that algorithm.

See [Knuth-TAOCP3A].

INPUT:

• n – (integer or Infinity) – size of the ground set
• t – (integer) – size of the subsets

EXAMPLES:

sage: from sage.combinat.gray_codes import combinations
sage: b = [1, 1, 1, 0, 0]
sage: for i,j in combinations(5,3):
....:     b[i] = 0; b[j] = 1
....:     print b
[1, 0, 1, 1, 0]
[0, 1, 1, 1, 0]
[1, 1, 0, 1, 0]
[1, 0, 0, 1, 1]
[0, 1, 0, 1, 1]
[0, 0, 1, 1, 1]
[1, 0, 1, 0, 1]
[0, 1, 1, 0, 1]
[1, 1, 0, 0, 1]

sage: s = set([0,1])
sage: for i,j in combinations(4,2):
....:     s.remove(i)
....:     print s
set([1, 2])
set([0, 2])
set([2, 3])
set([1, 3])
set([0, 3])


Note that n can be infinity:

sage: c = combinations(Infinity,4)
sage: s = set([0,1,2,3])
sage: for _ in xrange(10):
....:     i,j = next(c)
....:     print s
set([0, 1, 3, 4])
set([1, 2, 3, 4])
set([0, 2, 3, 4])
set([0, 1, 2, 4])
set([0, 1, 4, 5])
set([1, 2, 4, 5])
set([0, 2, 4, 5])
set([2, 3, 4, 5])
set([1, 3, 4, 5])
set([0, 3, 4, 5])
sage: for _ in xrange(1000):
....:     i,j = next(c)
sage: print s
set([0, 4, 13, 14])


TESTS:

sage: def check_sets_from_iter(n,k):
....:     l = []
....:     s = set(range(k))
....:     l.append(frozenset(s))
....:     for i,j in combinations(n,k):
....:         s.remove(i)
....:         assert len(s) == k
....:         l.append(frozenset(s))
....:     assert len(set(l)) == binomial(n,k)
sage: check_sets_from_iter(9,5)
sage: check_sets_from_iter(8,5)
sage: check_sets_from_iter(5,6)
Traceback (most recent call last):
...
AssertionError: t(=6) must be >=0 and <=n(=5)

sage.combinat.gray_codes.product(m)

Iterator over the switch for the iteration of the product $$[m_0] \times [m_1] \ldots \times [m_k]$$.

The iterator return at each step a pair (p,i) which corresponds to the modification to perform to get the next element. More precisely, one has to apply the increment i at the position p. By construction, the increment is either +1 or -1.

This is algorithm H in [Knuth-TAOCP2A]: loopless reflected mixed-radix Gray generation.

INPUT:

• m – a list or tuple of positive integers that correspond to the size of the sets in the product

EXAMPLES:

sage: from sage.combinat.gray_codes import product
sage: l = [0,0,0]
sage: for p,i in product([3,3,3]):
....:     l[p] += i
....:     print l
[1, 0, 0]
[2, 0, 0]
[2, 1, 0]
[1, 1, 0]
[0, 1, 0]
[0, 2, 0]
[1, 2, 0]
[2, 2, 0]
[2, 2, 1]
[1, 2, 1]
[0, 2, 1]
[0, 1, 1]
[1, 1, 1]
[2, 1, 1]
[2, 0, 1]
[1, 0, 1]
[0, 0, 1]
[0, 0, 2]
[1, 0, 2]
[2, 0, 2]
[2, 1, 2]
[1, 1, 2]
[0, 1, 2]
[0, 2, 2]
[1, 2, 2]
[2, 2, 2]
sage: l = [0,0]
sage: for i,j in product([2,1]):
....:     l[i] += j
....:     print l
[1, 0]


TESTS:

sage: for t in [[2,2,2],[2,1,2],[3,2,1],[2,1,3]]:
....:     assert sum(1 for _ in product(t)) == prod(t)-1