# Gray Code¶

class sympy.combinatorics.graycode.GrayCode(n, *args, **kw_args)[source]

A Gray code is essentially a Hamiltonian walk on a n-dimensional cube with edge length of one. The vertices of the cube are represented by vectors whose values are binary. The Hamilton walk visits each vertex exactly once. The Gray code for a 3d cube is [‘000’,’100’,’110’,’010’,’011’,’111’,’101’, ‘001’].

A Gray code solves the problem of sequentially generating all possible subsets of n objects in such a way that each subset is obtained from the previous one by either deleting or adding a single object. In the above example, 1 indicates that the object is present, and 0 indicates that its absent.

Gray codes have applications in statistics as well when we want to compute various statistics related to subsets in an efficient manner.

Examples

>>> from sympy.combinatorics.graycode import GrayCode
>>> a = GrayCode(3)
>>> list(a.generate_gray())
['000', '001', '011', '010', '110', '111', '101', '100']
>>> a = GrayCode(4)
>>> list(a.generate_gray())
['0000', '0001', '0011', '0010', '0110', '0111', '0101', '0100',     '1100', '1101', '1111', '1110', '1010', '1011', '1001', '1000']


References

R34

Nijenhuis,A. and Wilf,H.S.(1978). Combinatorial Algorithms. Academic Press.

R35

Knuth, D. (2011). The Art of Computer Programming, Vol 4 Addison Wesley

property current

Returns the currently referenced Gray code as a bit string.

Examples

>>> from sympy.combinatorics.graycode import GrayCode
>>> GrayCode(3, start='100').current
'100'

generate_gray(**hints)[source]

Generates the sequence of bit vectors of a Gray Code.

Examples

>>> from sympy.combinatorics.graycode import GrayCode
>>> a = GrayCode(3)
>>> list(a.generate_gray())
['000', '001', '011', '010', '110', '111', '101', '100']
>>> list(a.generate_gray(start='011'))
['011', '010', '110', '111', '101', '100']
>>> list(a.generate_gray(rank=4))
['110', '111', '101', '100']


References

R36

Knuth, D. (2011). The Art of Computer Programming, Vol 4, Addison Wesley

property n

Returns the dimension of the Gray code.

Examples

>>> from sympy.combinatorics.graycode import GrayCode
>>> a = GrayCode(5)
>>> a.n
5

next(delta=1)[source]

Returns the Gray code a distance delta (default = 1) from the current value in canonical order.

Examples

>>> from sympy.combinatorics.graycode import GrayCode
>>> a = GrayCode(3, start='110')
>>> a.next().current
'111'
>>> a.next(-1).current
'010'

property rank

Ranks the Gray code.

A ranking algorithm determines the position (or rank) of a combinatorial object among all the objects w.r.t. a given order. For example, the 4 bit binary reflected Gray code (BRGC) ‘0101’ has a rank of 6 as it appears in the 6th position in the canonical ordering of the family of 4 bit Gray codes.

Examples

>>> from sympy.combinatorics.graycode import GrayCode
>>> a = GrayCode(3)
>>> list(a.generate_gray())
['000', '001', '011', '010', '110', '111', '101', '100']
>>> GrayCode(3, start='100').rank
7
>>> GrayCode(3, rank=7).current
'100'


References

R37

http://statweb.stanford.edu/~susan/courses/s208/node12.html

property selections

Returns the number of bit vectors in the Gray code.

Examples

>>> from sympy.combinatorics.graycode import GrayCode
>>> a = GrayCode(3)
>>> a.selections
8

skip()[source]

Skips the bit generation.

Examples

>>> from sympy.combinatorics.graycode import GrayCode
>>> a = GrayCode(3)
>>> for i in a.generate_gray():
...     if i == '010':
...         a.skip()
...     print(i)
...
000
001
011
010
111
101
100

classmethod unrank(n, rank)[source]

Unranks an n-bit sized Gray code of rank k. This method exists so that a derivative GrayCode class can define its own code of a given rank.

The string here is generated in reverse order to allow for tail-call optimization.

Examples

>>> from sympy.combinatorics.graycode import GrayCode
>>> GrayCode(5, rank=3).current
'00010'
>>> GrayCode.unrank(5, 3)
'00010'

graycode.random_bitstring()[source]

Generates a random bitlist of length n.

Examples

>>> from sympy.combinatorics.graycode import random_bitstring
>>> random_bitstring(3)
100

graycode.gray_to_bin()[source]

Convert from Gray coding to binary coding.

We assume big endian encoding.

Examples

>>> from sympy.combinatorics.graycode import gray_to_bin
>>> gray_to_bin('100')
'111'

graycode.bin_to_gray()[source]

Convert from binary coding to gray coding.

We assume big endian encoding.

Examples

>>> from sympy.combinatorics.graycode import bin_to_gray
>>> bin_to_gray('111')
'100'

graycode.get_subset_from_bitstring(bitstring)[source]

Gets the subset defined by the bitstring.

Examples

>>> from sympy.combinatorics.graycode import get_subset_from_bitstring
>>> get_subset_from_bitstring(['a', 'b', 'c', 'd'], '0011')
['c', 'd']
>>> get_subset_from_bitstring(['c', 'a', 'c', 'c'], '1100')
['c', 'a']

graycode.graycode_subsets()[source]

Generates the subsets as enumerated by a Gray code.

Examples

>>> from sympy.combinatorics.graycode import graycode_subsets
>>> list(graycode_subsets(['a', 'b', 'c']))
[[], ['c'], ['b', 'c'], ['b'], ['a', 'b'], ['a', 'b', 'c'],     ['a', 'c'], ['a']]
>>> list(graycode_subsets(['a', 'b', 'c', 'c']))
[[], ['c'], ['c', 'c'], ['c'], ['b', 'c'], ['b', 'c', 'c'],     ['b', 'c'], ['b'], ['a', 'b'], ['a', 'b', 'c'], ['a', 'b', 'c', 'c'],     ['a', 'b', 'c'], ['a', 'c'], ['a', 'c', 'c'], ['a', 'c'], ['a']]