Hide keyboard shortcuts

Hot-keys on this page

r m x p   toggle line displays

j k   next/prev highlighted chunk

0   (zero) top of page

1   (one) first highlighted chunk

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

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

357

358

359

360

361

362

363

364

365

366

367

368

369

370

371

372

373

374

375

376

377

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

396

397

398

399

400

401

402

403

404

405

406

407

408

409

410

411

412

413

414

from __future__ import print_function, division 

 

from sympy.core import Basic 

from sympy.core.compatibility import range 

 

import random 

 

 

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 

 

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] 

 

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 

 

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 

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)