from __future__ import print_function, division
from sympy.core import Basic
from sympy.core.compatibility import range
import random
[docs]class GrayCode(Basic):
"""
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.
References:
[1] Nijenhuis,A. and Wilf,H.S.(1978).
Combinatorial Algorithms. Academic Press.
[2] Knuth, D. (2011). The Art of Computer Programming, Vol 4
Addison Wesley
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']
"""
_skip = False
_current = 0
_rank = None
def __new__(cls, n, *args, **kw_args):
"""
Default constructor.
It takes a single argument ``n`` which gives the dimension of the Gray
code. The starting Gray code string (``start``) or the starting ``rank``
may also be given; the default is to start at rank = 0 ('0...0').
Examples
========
>>> from sympy.combinatorics.graycode import GrayCode
>>> a = GrayCode(3)
>>> a
GrayCode(3)
>>> a.n
3
>>> a = GrayCode(3, start='100')
>>> a.current
'100'
>>> a = GrayCode(4, rank=4)
>>> a.current
'0110'
>>> a.rank
4
"""
if n < 1 or int(n) != n:
raise ValueError(
'Gray code dimension must be a positive integer, not %i' % n)
n = int(n)
args = (n,) + args
obj = Basic.__new__(cls, *args)
if 'start' in kw_args:
obj._current = kw_args["start"]
if len(obj._current) > n:
raise ValueError('Gray code start has length %i but '
'should not be greater than %i' % (len(obj._current), n))
elif 'rank' in kw_args:
if int(kw_args["rank"]) != kw_args["rank"]:
raise ValueError('Gray code rank must be a positive integer, '
'not %i' % kw_args["rank"])
obj._rank = int(kw_args["rank"]) % obj.selections
obj._current = obj.unrank(n, obj._rank)
return obj
[docs] def next(self, delta=1):
"""
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'
"""
return GrayCode(self.n, rank=(self.rank + delta) % self.selections)
@property
def selections(self):
"""
Returns the number of bit vectors in the Gray code.
Examples
========
>>> from sympy.combinatorics.graycode import GrayCode
>>> a = GrayCode(3)
>>> a.selections
8
"""
return 2**self.n
@property
def n(self):
"""
Returns the dimension of the Gray code.
Examples
========
>>> from sympy.combinatorics.graycode import GrayCode
>>> a = GrayCode(5)
>>> a.n
5
"""
return self.args[0]
[docs] def generate_gray(self, **hints):
"""
Generates the sequence of bit vectors of a Gray Code.
[1] Knuth, D. (2011). The Art of Computer Programming,
Vol 4, Addison Wesley
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']
See Also
========
skip
"""
bits = self.n
start = None
if "start" in hints:
start = hints["start"]
elif "rank" in hints:
start = GrayCode.unrank(self.n, hints["rank"])
if start is not None:
self._current = start
current = self.current
graycode_bin = gray_to_bin(current)
if len(graycode_bin) > self.n:
raise ValueError('Gray code start has length %i but should '
'not be greater than %i' % (len(graycode_bin), bits))
self._current = int(current, 2)
graycode_int = int(''.join(graycode_bin), 2)
for i in range(graycode_int, 1 << bits):
if self._skip:
self._skip = False
else:
yield self.current
bbtc = (i ^ (i + 1))
gbtc = (bbtc ^ (bbtc >> 1))
self._current = (self._current ^ gbtc)
self._current = 0
[docs] def skip(self):
"""
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
See Also
========
generate_gray
"""
self._skip = True
@property
def rank(self):
"""
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.
References:
[1] http://statweb.stanford.edu/~susan/courses/s208/node12.html
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'
See Also
========
unrank
"""
if self._rank is None:
self._rank = int(gray_to_bin(self.current), 2)
return self._rank
@property
def current(self):
"""
Returns the currently referenced Gray code as a bit string.
Examples
========
>>> from sympy.combinatorics.graycode import GrayCode
>>> GrayCode(3, start='100').current
'100'
"""
rv = self._current or '0'
if type(rv) is not str:
rv = bin(rv)[2:]
return rv.rjust(self.n, '0')
@classmethod
[docs] def unrank(self, n, rank):
"""
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'
See Also
========
rank
"""
def _unrank(k, n):
if n == 1:
return str(k % 2)
m = 2**(n - 1)
if k < m:
return '0' + _unrank(k, n - 1)
return '1' + _unrank(m - (k % m) - 1, n - 1)
return _unrank(rank, n)
def random_bitstring(n):
"""
Generates a random bitlist of length n.
Examples
========
>>> from sympy.combinatorics.graycode import random_bitstring
>>> random_bitstring(3) # doctest: +SKIP
100
"""
return ''.join([random.choice('01') for i in range(n)])
def gray_to_bin(bin_list):
"""
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'
See Also
========
bin_to_gray
"""
b = [bin_list[0]]
for i in range(1, len(bin_list)):
b += str(int(b[i - 1] != bin_list[i]))
return ''.join(b)
def bin_to_gray(bin_list):
"""
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'
See Also
========
gray_to_bin
"""
b = [bin_list[0]]
for i in range(0, len(bin_list) - 1):
b += str(int(bin_list[i]) ^ int(b[i - 1]))
return ''.join(b)
def get_subset_from_bitstring(super_set, bitstring):
"""
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']
See Also
========
graycode_subsets
"""
if len(super_set) != len(bitstring):
raise ValueError("The sizes of the lists are not equal")
return [super_set[i] for i, j in enumerate(bitstring)
if bitstring[i] == '1']
def graycode_subsets(gray_code_set):
"""
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']]
See Also
========
get_subset_from_bitstring
"""
for bitstring in list(GrayCode(len(gray_code_set)).generate_gray()):
yield get_subset_from_bitstring(gray_code_set, bitstring)
