Coverage for moptipy/examples/bitstrings/ising2d.py: 50%

1"""

2The two-dimensional Ising problem.

4The two-dimensional Ising problem describes a torus. For each bit that differs

5from its right-side neighboring bit, a penalty of 1 is incurred. For each bit

6differing from its above neighbor, another penalty of 1 is added.

7The optimum is a bit string of either all ones or all zeros.

8The optimal objective value is 0.

9The worst possible objective value is `2*n`.

111. Simon Fischer. A Polynomial Upper Bound for a Mutation-Based Algorithm on

12 the Two-Dimensional Ising Model. In Kalyanmoy Deb, editor, Genetic and

13 Evolutionary Computation Conference, June 26-30, 2004, Seattle, WA, USA,

14 Proceedings, Part I, Lecture Notes in Computer Science (LNCS), volume 3102,

15 Springer Berlin, Heidelberg. pages 1100-1112.

16 doi: https://doi.org/10.1007/978-3-540-24854-5_108.

172. Carola Doerr and Furong Ye and Naama Horesh and Hao Wang and Ofer M. Shir

18 and Thomas Bäck. Benchmarking Discrete Optimization Heuristics with

19 IOHprofiler. Applied Soft Computing 88:106027, March 2020,

20 doi: https://doi.org/10.1016/j.asoc.2019.106027.

213. Thomas Weise, Zhize Wu, Xinlu Li, Yan Chen, and Jörg Lässig. Frequency

22 Fitness Assignment: Optimization without Bias for Good Solutions can be

23 Efficient. *IEEE Transactions on Evolutionary Computation (TEVC)*.

24 27(4):980-992. August 2023.

25 doi: https://doi.org/10.1109/TEVC.2022.3191698

27This is code is part of the research work of Mr. Jiazheng ZENG (曾嘉政),

28a Master's student at the Institute of Applied Optimization

29(应用优化研究所) of the School of Artificial

30Intelligence and Big Data (人工智能与大数据学院) at

31Hefei University (合肥大学) in

32Hefei, Anhui, China (中国安徽省合肥市) under the supervision of

33Prof. Dr. Thomas Weise (汤卫思教授).

34"""

35from typing import Callable, Final, Iterator, cast

37import numba # type: ignore

38import numpy as np

40from moptipy.examples.bitstrings.bitstring_problem import (

41 SquareBitStringProblem,

42)

45@numba.njit(nogil=True, cache=True)

46def ising2d(x: np.ndarray, k: int) -> int:

47 """

48 Calculate the two-dimensional Ising model.

50 :param x: the flat numpy array representing the bitstring

51 :param k: the side length of the square grid (i.e., `sqrt(len(x)))`

52 :return: the two-dimensional Ising objective

54 >>> test_k = 2

55 >>> y = np.array([False] * test_k * test_k)

56 >>> ising2d(y, test_k)

57 0

59 >>> y.fill(True)

60 >>> ising2d(y, test_k)

61 0

63 >>> y[2] = not y[2]

64 >>> ising2d(y, test_k)

65 4

67 >>> ising2d(np.array([True, False, False, True]), 2)

68 8

70 >>> test_k = 3

71 >>> y = np.array([False] * test_k * test_k)

72 >>> ising2d(y, test_k)

73 0

75 >>> y.fill(True)

76 >>> ising2d(y, test_k)

77 0

79 >>> y[4] = not y[4]

80 >>> ising2d(y, test_k)

81 4

83 >>> y[5] = not y[5]

84 >>> ising2d(y, test_k)

85 6

87 >>> y[3] = not y[3]

88 >>> ising2d(y, test_k)

89 6

91 >>> test_k = 5

92 >>> y = np.array([False] * test_k * test_k)

93 >>> ising2d(y, test_k)

94 0

96 >>> y.fill(True)

97 >>> ising2d(y, test_k)

98 0

100 >>> y[5] = not y[5]

101 >>> ising2d(y, test_k)

102 4

103

104 >>> y[6] = not y[6]

105 >>> ising2d(y, test_k)

106 6

107

108 >>> y[7] = not y[7]

109 >>> ising2d(y, test_k)

110 8

111

112 >>> y[6] = not y[6]

113 >>> ising2d(y, test_k)

114 8

115

116 >>> y[10] = not y[10]

117 >>> ising2d(y, test_k)

118 10

119

120 >>> y[12] = not y[12]

121 >>> ising2d(y, test_k)

122 12

123

124 >>> y[11] = not y[11]

125 >>> ising2d(y, test_k)

126 12

127

128 >>> y[6] = not y[6]

129 >>> ising2d(y, test_k)

130 10

131

132 # 16 bits, torus width = 4, and 2 true bits

133 >>> ising2d(np.array([0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0]), 4)

134 8

135

136 # 16 bits, torus width = 4, and 3 true bits

137 >>> ising2d(np.array([0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0]), 4)

138 12

139

140 # 16 bits, torus width = 4, and 3 true bits

141 >>> ising2d(np.array([0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0]), 4)

142 10

143

144 # 16 bits, torus width = 4, and 1 true bit

145 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0]), 4)

146 4

147

148 # 16 bits, torus width = 4, and 4 true bits

149 >>> ising2d(np.array([0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0]), 4)

150 10

151

152 # 16 bits, torus width = 4, and 4 true bits

153 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1]), 4)

154 10

155

156 # 16 bits, torus width = 4, and 4 true bits

157 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0]), 4)

158 12

159

160 # 16 bits, torus width = 4, and 4 true bits

161 >>> ising2d(np.array([0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0]), 4)

162 12

163

164 # 16 bits, torus width = 4, and 10 true bits

165 >>> ising2d(np.array([0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1]), 4)

166 14

167

168 # 16 bits, torus width = 4, and 8 true bits

169 >>> ising2d(np.array([1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0]), 4)

170 22

171

172 # 16 bits, torus width = 4, and 12 true bits

173 >>> ising2d(np.array([1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0]), 4)

174 12

175

176 # 16 bits, torus width = 4, and 13 true bits

177 >>> ising2d(np.array([0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1]), 4)

178 12

179

180 # 16 bits, torus width = 4, and 0 true bits

181 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), 4)

182 0

183

184 # 16 bits, torus width = 4, and 16 true bits

185 >>> ising2d(np.array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]), 4)

186 0

187

188 # 25 bits, torus width = 5, and 1 true bit

189 >>> ising2d(np.array([0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

190 ... 0, 0, 0, 0, 0, 0, 0, 0]), 5)

191 4

192

193 # 25 bits, torus width = 5, and 1 true bit

194 >>> ising2d(np.array([0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

195 ... 0, 0, 0, 0, 0, 0, 0, 0]), 5)

196 4

197

198 # 25 bits, torus width = 5, and 1 true bit

199 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,

200 ... 0, 0, 0, 0, 0, 0, 0, 0]), 5)

201 4

202

203 # 25 bits, torus width = 5, and 4 true bits

204 >>> ising2d(np.array([1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,

205 ... 0, 1, 0, 1, 0, 0, 0, 0]), 5)

206 12

207

208 # 25 bits, torus width = 5, and 5 true bits

209 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,

210 ... 1, 0, 1, 0, 0, 1, 0, 0]), 5)

211 16

212

213 # 25 bits, torus width = 5, and 5 true bits

214 >>> ising2d(np.array([0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0,

215 ... 1, 0, 0, 0, 0, 0, 0, 1]), 5)

216 16

217

218 # 25 bits, torus width = 5, and 5 true bits

219 >>> ising2d(np.array([0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0,

220 ... 0, 0, 0, 0, 0, 0, 0, 1]), 5)

221 18

222

223 # 25 bits, torus width = 5, and 5 true bits

224 >>> ising2d(np.array([0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0,

225 ... 0, 0, 0, 0, 0, 0, 0, 1]), 5)

226 20

227

228 # 25 bits, torus width = 5, and 13 true bits

229 >>> ising2d(np.array([1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1,

230 ... 1, 0, 1, 0, 1, 0, 0, 1]), 5)

231 26

232

233 # 25 bits, torus width = 5, and 21 true bits

234 >>> ising2d(np.array([1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1,

235 ... 1, 1, 1, 1, 1, 1, 1, 1]), 5)

236 12

237

238 # 25 bits, torus width = 5, and 24 true bits

239 >>> ising2d(np.array([1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

240 ... 1, 1, 1, 1, 1, 1, 1, 1]), 5)

241 4

242

243 # 25 bits, torus width = 5, and 12 true bits

244 >>> ising2d(np.array([1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0,

245 ... 0, 1, 0, 0, 1, 0, 1, 0]), 5)

246 24

247

248 # 25 bits, torus width = 5, and 0 true bits

249 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

250 ... 0, 0, 0, 0, 0, 0, 0, 0]), 5)

251 0

252

253 # 25 bits, torus width = 5, and 25 true bits

254 >>> ising2d(np.array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

255 ... 1, 1, 1, 1, 1, 1, 1, 1]), 5)

256 0

257

258 # 36 bits, torus width = 6, and 3 true bits

259 >>> ising2d(np.array([0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,

260 ... 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), 6)

261 12

262

263 # 36 bits, torus width = 6, and 4 true bits

264 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

265 ... 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]), 6)

266 12

267

268 # 36 bits, torus width = 6, and 2 true bits

269 >>> ising2d(np.array([0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,

270 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), 6)

271 8

272

273 # 36 bits, torus width = 6, and 1 true bit

274 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

275 ... 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), 6)

276 4

277

278 # 36 bits, torus width = 6, and 6 true bits

279 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,

280 ... 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0]), 6)

281 18

282

283 # 36 bits, torus width = 6, and 6 true bits

284 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0,

285 ... 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0]), 6)

286 18

287

288 # 36 bits, torus width = 6, and 6 true bits

289 >>> ising2d(np.array([0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0,

290 ... 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0]), 6)

291 18

292

293 # 36 bits, torus width = 6, and 6 true bits

294 >>> ising2d(np.array([0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0,

295 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0]), 6)

296 18

297

298 # 36 bits, torus width = 6, and 12 true bits

299 >>> ising2d(np.array([0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0,

300 ... 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0]), 6)

301 28

302

303 # 36 bits, torus width = 6, and 15 true bits

304 >>> ising2d(np.array([1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0,

305 ... 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1]), 6)

306 34

307

308 # 36 bits, torus width = 6, and 26 true bits

309 >>> ising2d(np.array([1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1,

310 ... 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1]), 6)

311 30

312

313 # 36 bits, torus width = 6, and 20 true bits

314 >>> ising2d(np.array([0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1,

315 ... 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0]), 6)

316 32

317

318 # 36 bits, torus width = 6, and 0 true bits

319 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

320 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), 6)

321 0

322

323 # 36 bits, torus width = 6, and 36 true bits

324 >>> ising2d(np.array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

325 ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]), 6)

326 0

327

328 # 49 bits, torus width = 7, and 6 true bits

329 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,

330 ... 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

331 ... 0, 0, 0, 0, 1, 0, 0, 0, 1, 0]), 7)

332 24

333

334 # 49 bits, torus width = 7, and 4 true bits

335 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,

336 ... 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,

337 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), 7)

338 16

339

340 # 49 bits, torus width = 7, and 2 true bits

341 >>> ising2d(np.array([0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,

342 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

343 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), 7)

344 8

345

346 # 49 bits, torus width = 7, and 1 true bit

347 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

348 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

349 ... 0, 0, 0, 0, 0, 0, 1, 0, 0, 0]), 7)

350 4

351

352 # 49 bits, torus width = 7, and 7 true bits

353 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,

354 ... 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0,

355 ... 0, 1, 0, 0, 0, 0, 0, 1, 0, 0]), 7)

356 24

357

358 # 49 bits, torus width = 7, and 7 true bits

359 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,

360 ... 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0,

361 ... 1, 0, 0, 0, 0, 1, 0, 0, 0, 0]), 7)

362 26

363

364 # 49 bits, torus width = 7, and 7 true bits

365 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,

366 ... 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1,

367 ... 0, 0, 1, 0, 1, 0, 0, 0, 1, 0]), 7)

368 28

369

370 # 49 bits, torus width = 7, and 7 true bits

371 >>> ising2d(np.array([0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,

372 ... 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0,

373 ... 0, 0, 1, 0, 0, 0, 0, 0, 0, 0]), 7)

374 26

375

376 # 49 bits, torus width = 7, and 14 true bits

377 >>> ising2d(np.array([0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1,

378 ... 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0,

379 ... 0, 0, 0, 0, 0, 1, 0, 0, 0, 0]), 7)

380 36

381

382 # 49 bits, torus width = 7, and 12 true bits

383 >>> ising2d(np.array([0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0,

384 ... 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,

385 ... 1, 0, 0, 1, 0, 0, 1, 0, 0, 0]), 7)

386 38

387

388 # 49 bits, torus width = 7, and 15 true bits

389 >>> ising2d(np.array([0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0,

390 ... 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0,

391 ... 1, 1, 1, 0, 0, 1, 0, 0, 1, 0]), 7)

392 42

393

394 # 49 bits, torus width = 7, and 29 true bits

395 >>> ising2d(np.array([1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1,

396 ... 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0,

397 ... 0, 1, 1, 1, 1, 0, 1, 1, 1, 1]), 7)

398 48

399

400 # 49 bits, torus width = 7, and 0 true bits

401 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

402 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

403 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), 7)

404 0

405

406 # 49 bits, torus width = 7, and 49 true bits

407 >>> ising2d(np.array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

408 ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

409 ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]), 7)

410 0

411

412 # 64 bits, torus width = 8, and 5 true bits

413 >>> ising2d(np.array([0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

414 ... 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

415 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0,

416 ... 0, 0, 0]), 8)

417 20

418

419 # 64 bits, torus width = 8, and 7 true bits

420 >>> ising2d(np.array([0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

421 ... 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1,

422 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0,

423 ... 0, 0, 0]), 8)

424 26

425

426 # 64 bits, torus width = 8, and 2 true bits

427 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

428 ... 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0,

429 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

430 ... 0, 0, 0]), 8)

431 8

432

433 # 64 bits, torus width = 8, and 4 true bits

434 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,

435 ... 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0,

436 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

437 ... 0, 0, 0]), 8)

438 14

439

440 # 64 bits, torus width = 8, and 8 true bits

441 >>> ising2d(np.array([0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

442 ... 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,

443 ... 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,

444 ... 1, 0, 0]), 8)

445 30

446

447 # 64 bits, torus width = 8, and 8 true bits

448 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,

449 ... 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

450 ... 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0,

451 ... 0, 0, 0]), 8)

452 28

453

454 # 64 bits, torus width = 8, and 8 true bits

455 >>> ising2d(np.array([0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,

456 ... 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,

457 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0,

458 ... 0, 1, 0]), 8)

459 30

460

461 # 64 bits, torus width = 8, and 8 true bits

462 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1,

463 ... 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,

464 ... 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

465 ... 0, 0, 0]), 8)

466 26

467

468 # 64 bits, torus width = 8, and 37 true bits

469 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1,

470 ... 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1,

471 ... 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0,

472 ... 1, 1, 0]), 8)

473 58

474

475 # 64 bits, torus width = 8, and 36 true bits

476 >>> ising2d(np.array([0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1,

477 ... 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0,

478 ... 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1,

479 ... 1, 1, 1]), 8)

480 62

481

482 # 64 bits, torus width = 8, and 55 true bits

483 >>> ising2d(np.array([1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1,

484 ... 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0,

485 ... 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1,

486 ... 1, 1, 1]), 8)

487 32

488

489 # 64 bits, torus width = 8, and 38 true bits

490 >>> ising2d(np.array([0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1,

491 ... 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1,

492 ... 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1,

493 ... 1, 0, 1]), 8)

494 68

495

496 # 64 bits, torus width = 8, and 0 true bits

497 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

498 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

499 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

500 ... 0, 0, 0]), 8)

501 0

502

503 # 64 bits, torus width = 8, and 64 true bits

504 >>> ising2d(np.array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

505 ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

506 ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

507 ... 1, 1, 1]), 8)

508 0

509

510 # 81 bits, torus width = 9, and 1 true bit

511 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

512 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,

513 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

514 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), 9)

515 4

516

517 # 81 bits, torus width = 9, and 5 true bits

518 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

519 ... 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,

520 ... 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,

521 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), 9)

522 20

523

524 # 81 bits, torus width = 9, and 1 true bit

525 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

526 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

527 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

528 ... 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), 9)

529 4

530

531 # 81 bits, torus width = 9, and 3 true bits

532 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,

533 ... 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

534 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,

535 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), 9)

536 12

537

538 # 81 bits, torus width = 9, and 9 true bits

539 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0,

540 ... 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0,

541 ... 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

542 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0]), 9)

543 30

544

545 # 81 bits, torus width = 9, and 9 true bits

546 >>> ising2d(np.array([0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0,

547 ... 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,

548 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

549 ... 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), 9)

550 32

551

552 # 81 bits, torus width = 9, and 9 true bits

553 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0,

554 ... 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,

555 ... 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0,

556 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]), 9)

557 32

558

559 # 81 bits, torus width = 9, and 9 true bits

560 >>> ising2d(np.array([0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,

561 ... 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0,

562 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0,

563 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0]), 9)

564 36

565

566 # 81 bits, torus width = 9, and 11 true bits

567 >>> ising2d(np.array([0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,

568 ... 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,

569 ... 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0,

570 ... 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]), 9)

571 40

572

573 # 81 bits, torus width = 9, and 40 true bits

574 >>> ising2d(np.array([1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0,

575 ... 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1,

576 ... 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1,

577 ... 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0]), 9)

578 88

579

580 # 81 bits, torus width = 9, and 35 true bits

581 >>> ising2d(np.array([0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0,

582 ... 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0,

583 ... 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0,

584 ... 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1]), 9)

585 70

586

587 # 81 bits, torus width = 9, and 79 true bits

588 >>> ising2d(np.array([1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1,

589 ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

590 ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1,

591 ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]), 9)

592 8

593

594 # 81 bits, torus width = 9, and 0 true bits

595 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

596 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

597 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

598 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), 9)

599 0

600

601 # 81 bits, torus width = 9, and 81 true bits

602 >>> ising2d(np.array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

603 ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

604 ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

605 ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]), 9)

606 0

607

608 # 100 bits, torus width = 10, and 3 true bits

609 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,

610 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

611 ... 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

612 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

613 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0]), 10)

614 12

615

616 # 100 bits, torus width = 10, and 5 true bits

617 >>> ising2d(np.array([0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

618 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,

619 ... 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

620 ... 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,

621 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), 10)

622 20

623

624 # 100 bits, torus width = 10, and 2 true bits

625 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

626 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

627 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

628 ... 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

629 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), 10)

630 8

631

632 # 100 bits, torus width = 10, and 1 true bit

633 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

634 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

635 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

636 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,

637 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), 10)

638 4

639

640 # 100 bits, torus width = 10, and 10 true bits

641 >>> ising2d(np.array([0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,

642 ... 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

643 ... 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

644 ... 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,

645 ... 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0]), 10)

646 40

647

648 # 100 bits, torus width = 10, and 10 true bits

649 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

650 ... 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,

651 ... 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

652 ... 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1,

653 ... 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0]), 10)

654 38

655

656 # 100 bits, torus width = 10, and 10 true bits

657 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

658 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1,

659 ... 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1,

660 ... 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

661 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), 10)

662 28

663

664 # 100 bits, torus width = 10, and 10 true bits

665 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

666 ... 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,

667 ... 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

668 ... 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0,

669 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0]), 10)

670 36

671

672 # 100 bits, torus width = 10, and 91 true bits

673 >>> ising2d(np.array([1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1,

674 ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1,

675 ... 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1,

676 ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1,

677 ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]), 10)

678 32

679

680 # 100 bits, torus width = 10, and 32 true bits

681 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0,

682 ... 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1,

683 ... 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0,

684 ... 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1,

685 ... 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0]), 10)

686 68

687

688 # 100 bits, torus width = 10, and 89 true bits

689 >>> ising2d(np.array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

690 ... 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

691 ... 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

692 ... 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

693 ... 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0]), 10)

694 42

695

696 # 100 bits, torus width = 10, and 92 true bits

697 >>> ising2d(np.array([0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1,

698 ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1,

699 ... 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1,

700 ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1,

701 ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]), 10)

702 30

703

704 # 100 bits, torus width = 10, and 0 true bits

705 >>> ising2d(np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

706 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

707 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

708 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

709 ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), 10)

710 0

711

712 # 100 bits, torus width = 10, and 100 true bits

713 >>> ising2d(np.array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

714 ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

715 ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

716 ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

717 ... 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]), 10)

718 0

719 """

720 n: Final[int] = len(x) # Get the length of the bit string.

721 result: int = n + n # Start with worst-case result.

722

723 prev_row_start: int = n - k # The start of the bottom row.

724 cur_row_start: int = 0 # The start index of the first row.

725 for _ in range(k): # Iterate over the rows.

726 prev_j: int = k - 1 # The previous index = last element.

727 for j in range(k): # Iterate over the current row.

728 center = x[cur_row_start + j] # The current element.

729 if center == x[prev_row_start + j]: # One row above the same?

730 result -= 1 # If above element same, no penalty.

731 if center == x[cur_row_start + prev_j]: # Element to the right.

732 result -= 1 # If right element same, no penalty.

733 prev_j = j # Previous element <-- current element.

734 prev_row_start = cur_row_start # Previous row <-- current row.

735 cur_row_start += k # k elements per row.

736

737 return result

738

739

740class Ising2d(SquareBitStringProblem):

741 """

742 The two-dimensional Ising model.

743

744 >>> ising = Ising2d(16)

745 >>> ising.n

746 16

747 >>> ising.k

748 4

749 >>> ising.evaluate(np.array([False, False, False, False,

750 ... False, False, False, False,

751 ... False, False, False, False,

752 ... False, False, False, False]))

753 0

754 >>> ising.evaluate(np.array([False, False, False, False,

755 ... False, True, False, False,

756 ... False, False, False, False,

757 ... False, False, False, False]))

758 4

759 >>> ising.evaluate(np.array([False, False, False, False,

760 ... True, True, True, True,

761 ... False, False, False, False,

762 ... False, False, False, False]))

763 8

764 >>> ising.evaluate(np.array([False, True, False, False,

765 ... True, True, True, True,

766 ... False, True, False, False,

767 ... False, True, False, False]))

768 12

769 >>> ising.evaluate(np.array([False, True, False, False,

770 ... True, True, True, True,

771 ... False, True, False, False,

772 ... False, True, False, True]))

773 16

774 >>> ising.evaluate(np.array([False, True, False, False,

775 ... True, True, True, True,

776 ... False, True, False, False,

777 ... True, True, True, True]))

778 16

779 """

780

781 def evaluate(self, x: np.ndarray) -> int:

782 """

783 Evaluate a solution to the 2D Ising problem.

784

785 :param x: the bit string to evaluate

786 :returns: the value of the 2D Ising Model for the string

787 """

788 return ising2d(x, self.k)

789

790 def upper_bound(self) -> int:

791 """

792 Get the upper bound of the two-dimensional Ising model.

793

794 :return: twice the length of the bit string

795

796 >>> Ising2d(49).upper_bound()

797 98

798 """

799 return 2 * self.n

800

801 def __str__(self) -> str:

802 """

803 Get the name of the two-dimensional Ising model objective function.

804

805 :return: ising2d_ + length of string

806

807 >>> Ising2d(16)

808 ising2d_16

809 """

810 return f"ising2d_{self.n}"

811

812 @classmethod

813 def default_instances(

814 cls: type, scale_min: int = 2, scale_max: int = 100) \

815 -> Iterator[Callable[[], "Ising2d"]]:

816 """

817 Get the 56 default instances of the :class:`Ising2d` problem.

818

819 :param scale_min: the minimum permitted scale, by default `2`

820 :param scale_max: the maximum permitted scale, by default `100`

821 :returns: a sequence of default :class:`Ising2d` instances

822

823 >>> len(list(Ising2d.default_instances()))

824 9

825

826 >>> [x() for x in Ising2d.default_instances()]

827 [ising2d_4, ising2d_9, ising2d_16, ising2d_25, ising2d_36, \

828ising2d_49, ising2d_64, ising2d_81, ising2d_100]

829 """

830 return cast("Iterator[Callable[[], Ising2d]]",

831 super().default_instances( # type: ignore

832 scale_min, scale_max))

Coverage for moptipy / examples / bitstrings / ising2d.py: 50%

32 statements