Coverage for /pythoncovmergedfiles/medio/medio/usr/local/lib/python3.11/site-packages/networkx/algorithms/tree/mst.py: 15%

1""" 

2Algorithms for calculating min/max spanning trees/forests. 

3 

4""" 

5 

6from dataclasses import dataclass, field 

7from enum import Enum 

8from heapq import heappop, heappush 

9from itertools import count 

10from math import isnan 

11from operator import itemgetter 

12from queue import PriorityQueue 

13 

14import networkx as nx 

15from networkx.utils import UnionFind, not_implemented_for, py_random_state 

16 

17__all__ = [ 

18 "minimum_spanning_edges", 

19 "maximum_spanning_edges", 

20 "minimum_spanning_tree", 

21 "maximum_spanning_tree", 

22 "number_of_spanning_trees", 

23 "random_spanning_tree", 

24 "partition_spanning_tree", 

25 "EdgePartition", 

26 "SpanningTreeIterator", 

27] 

28 

29 

30class EdgePartition(Enum): 

31 """ 

32 An enum to store the state of an edge partition. The enum is written to the 

33 edges of a graph before being pasted to `kruskal_mst_edges`. Options are: 

34 

35 - EdgePartition.OPEN 

36 - EdgePartition.INCLUDED 

37 - EdgePartition.EXCLUDED 

38 """ 

39 

40 OPEN = 0 

41 INCLUDED = 1 

42 EXCLUDED = 2 

43 

44 

45@not_implemented_for("multigraph") 

46@nx._dispatchable(edge_attrs="weight", preserve_edge_attrs="data") 

47def boruvka_mst_edges( 

48 G, minimum=True, weight="weight", keys=False, data=True, ignore_nan=False 

49): 

50 """Iterate over edges of a Borůvka's algorithm min/max spanning tree. 

51 

52 Parameters 

53 ---------- 

54 G : NetworkX Graph 

55 The edges of `G` must have distinct weights, 

56 otherwise the edges may not form a tree. 

57 

58 minimum : bool (default: True) 

59 Find the minimum (True) or maximum (False) spanning tree. 

60 

61 weight : string (default: 'weight') 

62 The name of the edge attribute holding the edge weights. 

63 

64 keys : bool (default: True) 

65 This argument is ignored since this function is not 

66 implemented for multigraphs; it exists only for consistency 

67 with the other minimum spanning tree functions. 

68 

69 data : bool (default: True) 

70 Flag for whether to yield edge attribute dicts. 

71 If True, yield edges `(u, v, d)`, where `d` is the attribute dict. 

72 If False, yield edges `(u, v)`. 

73 

74 ignore_nan : bool (default: False) 

75 If a NaN is found as an edge weight normally an exception is raised. 

76 If `ignore_nan is True` then that edge is ignored instead. 

77 

78 """ 

79 # Initialize a forest, assuming initially that it is the discrete 

80 # partition of the nodes of the graph. 

81 forest = UnionFind(G) 

82 

83 def best_edge(component): 

84 """Returns the optimum (minimum or maximum) edge on the edge 

85 boundary of the given set of nodes. 

86 

87 A return value of ``None`` indicates an empty boundary. 

88 

89 """ 

90 sign = 1 if minimum else -1 

91 minwt = float("inf") 

92 boundary = None 

93 for e in nx.edge_boundary(G, component, data=True): 

94 wt = e[-1].get(weight, 1) * sign 

95 if isnan(wt): 

96 if ignore_nan: 

97 continue 

98 msg = f"NaN found as an edge weight. Edge {e}" 

99 raise ValueError(msg) 

100 if wt < minwt: 

101 minwt = wt 

102 boundary = e 

103 return boundary 

104 

105 # Determine the optimum edge in the edge boundary of each component 

106 # in the forest. 

107 best_edges = (best_edge(component) for component in forest.to_sets()) 

108 best_edges = [edge for edge in best_edges if edge is not None] 

109 # If each entry was ``None``, that means the graph was disconnected, 

110 # so we are done generating the forest. 

111 while best_edges: 

112 # Determine the optimum edge in the edge boundary of each 

113 # component in the forest. 

114 # 

115 # This must be a sequence, not an iterator. In this list, the 

116 # same edge may appear twice, in different orientations (but 

117 # that's okay, since a union operation will be called on the 

118 # endpoints the first time it is seen, but not the second time). 

119 # 

120 # Any ``None`` indicates that the edge boundary for that 

121 # component was empty, so that part of the forest has been 

122 # completed. 

123 # 

124 # TODO This can be parallelized, both in the outer loop over 

125 # each component in the forest and in the computation of the 

126 # minimum. (Same goes for the identical lines outside the loop.) 

127 best_edges = (best_edge(component) for component in forest.to_sets()) 

128 best_edges = [edge for edge in best_edges if edge is not None] 

129 # Join trees in the forest using the best edges, and yield that 

130 # edge, since it is part of the spanning tree. 

131 # 

132 # TODO This loop can be parallelized, to an extent (the union 

133 # operation must be atomic). 

134 for u, v, d in best_edges: 

135 if forest[u] != forest[v]: 

136 if data: 

137 yield u, v, d 

138 else: 

139 yield u, v 

140 forest.union(u, v) 

141 

142 

143@nx._dispatchable( 

144 edge_attrs={"weight": None, "partition": None}, preserve_edge_attrs="data" 

145) 

146def kruskal_mst_edges( 

147 G, minimum, weight="weight", keys=True, data=True, ignore_nan=False, partition=None 

148): 

149 """ 

150 Iterate over edge of a Kruskal's algorithm min/max spanning tree. 

151 

152 Parameters 

153 ---------- 

154 G : NetworkX Graph 

155 The graph holding the tree of interest. 

156 

157 minimum : bool (default: True) 

158 Find the minimum (True) or maximum (False) spanning tree. 

159 

160 weight : string (default: 'weight') 

161 The name of the edge attribute holding the edge weights. 

162 

163 keys : bool (default: True) 

164 If `G` is a multigraph, `keys` controls whether edge keys ar yielded. 

165 Otherwise `keys` is ignored. 

166 

167 data : bool (default: True) 

168 Flag for whether to yield edge attribute dicts. 

169 If True, yield edges `(u, v, d)`, where `d` is the attribute dict. 

170 If False, yield edges `(u, v)`. 

171 

172 ignore_nan : bool (default: False) 

173 If a NaN is found as an edge weight normally an exception is raised. 

174 If `ignore_nan is True` then that edge is ignored instead. 

175 

176 partition : string (default: None) 

177 The name of the edge attribute holding the partition data, if it exists. 

178 Partition data is written to the edges using the `EdgePartition` enum. 

179 If a partition exists, all included edges and none of the excluded edges 

180 will appear in the final tree. Open edges may or may not be used. 

181 

182 Yields 

183 ------ 

184 edge tuple 

185 The edges as discovered by Kruskal's method. Each edge can 

186 take the following forms: `(u, v)`, `(u, v, d)` or `(u, v, k, d)` 

187 depending on the `key` and `data` parameters 

188 """ 

189 subtrees = UnionFind() 

190 if G.is_multigraph(): 

191 edges = G.edges(keys=True, data=True) 

192 else: 

193 edges = G.edges(data=True) 

194 

195 # Sort the edges of the graph with respect to the partition data. 

196 # Edges are returned in the following order: 

197 

198 # * Included edges 

199 # * Open edges from smallest to largest weight 

200 # * Excluded edges 

201 included_edges = [] 

202 open_edges = [] 

203 for e in edges: 

204 d = e[-1] 

205 wt = d.get(weight, 1) 

206 if isnan(wt): 

207 if ignore_nan: 

208 continue 

209 raise ValueError(f"NaN found as an edge weight. Edge {e}") 

210 

211 edge = (wt,) + e 

212 if d.get(partition) == EdgePartition.INCLUDED: 

213 included_edges.append(edge) 

214 elif d.get(partition) == EdgePartition.EXCLUDED: 

215 continue 

216 else: 

217 open_edges.append(edge) 

218 

219 if minimum: 

220 sorted_open_edges = sorted(open_edges, key=itemgetter(0)) 

221 else: 

222 sorted_open_edges = sorted(open_edges, key=itemgetter(0), reverse=True) 

223 

224 # Condense the lists into one 

225 included_edges.extend(sorted_open_edges) 

226 sorted_edges = included_edges 

227 del open_edges, sorted_open_edges, included_edges 

228 

229 # Multigraphs need to handle edge keys in addition to edge data. 

230 if G.is_multigraph(): 

231 for wt, u, v, k, d in sorted_edges: 

232 if subtrees[u] != subtrees[v]: 

233 if keys: 

234 if data: 

235 yield u, v, k, d 

236 else: 

237 yield u, v, k 

238 else: 

239 if data: 

240 yield u, v, d 

241 else: 

242 yield u, v 

243 subtrees.union(u, v) 

244 else: 

245 for wt, u, v, d in sorted_edges: 

246 if subtrees[u] != subtrees[v]: 

247 if data: 

248 yield u, v, d 

249 else: 

250 yield u, v 

251 subtrees.union(u, v) 

252 

253 

254@nx._dispatchable(edge_attrs="weight", preserve_edge_attrs="data") 

255def prim_mst_edges(G, minimum, weight="weight", keys=True, data=True, ignore_nan=False): 

256 """Iterate over edges of Prim's algorithm min/max spanning tree. 

257 

258 Parameters 

259 ---------- 

260 G : NetworkX Graph 

261 The graph holding the tree of interest. 

262 

263 minimum : bool (default: True) 

264 Find the minimum (True) or maximum (False) spanning tree. 

265 

266 weight : string (default: 'weight') 

267 The name of the edge attribute holding the edge weights. 

268 

269 keys : bool (default: True) 

270 If `G` is a multigraph, `keys` controls whether edge keys ar yielded. 

271 Otherwise `keys` is ignored. 

272 

273 data : bool (default: True) 

274 Flag for whether to yield edge attribute dicts. 

275 If True, yield edges `(u, v, d)`, where `d` is the attribute dict. 

276 If False, yield edges `(u, v)`. 

277 

278 ignore_nan : bool (default: False) 

279 If a NaN is found as an edge weight normally an exception is raised. 

280 If `ignore_nan is True` then that edge is ignored instead. 

281 

282 """ 

283 is_multigraph = G.is_multigraph() 

284 

285 nodes = set(G) 

286 c = count() 

287 

288 sign = 1 if minimum else -1 

289 

290 while nodes: 

291 u = nodes.pop() 

292 frontier = [] 

293 visited = {u} 

294 if is_multigraph: 

295 for v, keydict in G.adj[u].items(): 

296 for k, d in keydict.items(): 

297 wt = d.get(weight, 1) * sign 

298 if isnan(wt): 

299 if ignore_nan: 

300 continue 

301 msg = f"NaN found as an edge weight. Edge {(u, v, k, d)}" 

302 raise ValueError(msg) 

303 heappush(frontier, (wt, next(c), u, v, k, d)) 

304 else: 

305 for v, d in G.adj[u].items(): 

306 wt = d.get(weight, 1) * sign 

307 if isnan(wt): 

308 if ignore_nan: 

309 continue 

310 msg = f"NaN found as an edge weight. Edge {(u, v, d)}" 

311 raise ValueError(msg) 

312 heappush(frontier, (wt, next(c), u, v, d)) 

313 while nodes and frontier: 

314 if is_multigraph: 

315 W, _, u, v, k, d = heappop(frontier) 

316 else: 

317 W, _, u, v, d = heappop(frontier) 

318 if v in visited or v not in nodes: 

319 continue 

320 # Multigraphs need to handle edge keys in addition to edge data. 

321 if is_multigraph and keys: 

322 if data: 

323 yield u, v, k, d 

324 else: 

325 yield u, v, k 

326 else: 

327 if data: 

328 yield u, v, d 

329 else: 

330 yield u, v 

331 # update frontier 

332 visited.add(v) 

333 nodes.discard(v) 

334 if is_multigraph: 

335 for w, keydict in G.adj[v].items(): 

336 if w in visited: 

337 continue 

338 for k2, d2 in keydict.items(): 

339 new_weight = d2.get(weight, 1) * sign 

340 if isnan(new_weight): 

341 if ignore_nan: 

342 continue 

343 msg = f"NaN found as an edge weight. Edge {(v, w, k2, d2)}" 

344 raise ValueError(msg) 

345 heappush(frontier, (new_weight, next(c), v, w, k2, d2)) 

346 else: 

347 for w, d2 in G.adj[v].items(): 

348 if w in visited: 

349 continue 

350 new_weight = d2.get(weight, 1) * sign 

351 if isnan(new_weight): 

352 if ignore_nan: 

353 continue 

354 msg = f"NaN found as an edge weight. Edge {(v, w, d2)}" 

355 raise ValueError(msg) 

356 heappush(frontier, (new_weight, next(c), v, w, d2)) 

357 

358 

359ALGORITHMS = { 

360 "boruvka": boruvka_mst_edges, 

361 "borůvka": boruvka_mst_edges, 

362 "kruskal": kruskal_mst_edges, 

363 "prim": prim_mst_edges, 

364} 

365 

366 

367@not_implemented_for("directed") 

368@nx._dispatchable(edge_attrs="weight", preserve_edge_attrs="data") 

369def minimum_spanning_edges( 

370 G, algorithm="kruskal", weight="weight", keys=True, data=True, ignore_nan=False 

371): 

372 """Generate edges in a minimum spanning forest of an undirected 

373 weighted graph. 

374 

375 A minimum spanning tree is a subgraph of the graph (a tree) 

376 with the minimum sum of edge weights. A spanning forest is a 

377 union of the spanning trees for each connected component of the graph. 

378 

379 Parameters 

380 ---------- 

381 G : undirected Graph 

382 An undirected graph. If `G` is connected, then the algorithm finds a 

383 spanning tree. Otherwise, a spanning forest is found. 

384 

385 algorithm : string 

386 The algorithm to use when finding a minimum spanning tree. Valid 

387 choices are 'kruskal', 'prim', or 'boruvka'. The default is 'kruskal'. 

388 

389 weight : string 

390 Edge data key to use for weight (default 'weight'). 

391 

392 keys : bool 

393 Whether to yield edge key in multigraphs in addition to the edge. 

394 If `G` is not a multigraph, this is ignored. 

395 

396 data : bool, optional 

397 If True yield the edge data along with the edge. 

398 

399 ignore_nan : bool (default: False) 

400 If a NaN is found as an edge weight normally an exception is raised. 

401 If `ignore_nan is True` then that edge is ignored instead. 

402 

403 Returns 

404 ------- 

405 edges : iterator 

406 An iterator over edges in a maximum spanning tree of `G`. 

407 Edges connecting nodes `u` and `v` are represented as tuples: 

408 `(u, v, k, d)` or `(u, v, k)` or `(u, v, d)` or `(u, v)` 

409 

410 If `G` is a multigraph, `keys` indicates whether the edge key `k` will 

411 be reported in the third position in the edge tuple. `data` indicates 

412 whether the edge datadict `d` will appear at the end of the edge tuple. 

413 

414 If `G` is not a multigraph, the tuples are `(u, v, d)` if `data` is True 

415 or `(u, v)` if `data` is False. 

416 

417 Examples 

418 -------- 

419 >>> from networkx.algorithms import tree 

420 

421 Find minimum spanning edges by Kruskal's algorithm 

422 

423 >>> G = nx.cycle_graph(4) 

424 >>> G.add_edge(0, 3, weight=2) 

425 >>> mst = tree.minimum_spanning_edges(G, algorithm="kruskal", data=False) 

426 >>> edgelist = list(mst) 

427 >>> sorted(sorted(e) for e in edgelist) 

428 [[0, 1], [1, 2], [2, 3]] 

429 

430 Find minimum spanning edges by Prim's algorithm 

431 

432 >>> G = nx.cycle_graph(4) 

433 >>> G.add_edge(0, 3, weight=2) 

434 >>> mst = tree.minimum_spanning_edges(G, algorithm="prim", data=False) 

435 >>> edgelist = list(mst) 

436 >>> sorted(sorted(e) for e in edgelist) 

437 [[0, 1], [1, 2], [2, 3]] 

438 

439 Notes 

440 ----- 

441 For Borůvka's algorithm, each edge must have a weight attribute, and 

442 each edge weight must be distinct. 

443 

444 For the other algorithms, if the graph edges do not have a weight 

445 attribute a default weight of 1 will be used. 

446 

447 Modified code from David Eppstein, April 2006 

448 http://www.ics.uci.edu/~eppstein/PADS/ 

449 

450 """ 

451 try: 

452 algo = ALGORITHMS[algorithm] 

453 except KeyError as err: 

454 msg = f"{algorithm} is not a valid choice for an algorithm." 

455 raise ValueError(msg) from err 

456 

457 return algo( 

458 G, minimum=True, weight=weight, keys=keys, data=data, ignore_nan=ignore_nan 

459 ) 

460 

461 

462@not_implemented_for("directed") 

463@nx._dispatchable(edge_attrs="weight", preserve_edge_attrs="data") 

464def maximum_spanning_edges( 

465 G, algorithm="kruskal", weight="weight", keys=True, data=True, ignore_nan=False 

466): 

467 """Generate edges in a maximum spanning forest of an undirected 

468 weighted graph. 

469 

470 A maximum spanning tree is a subgraph of the graph (a tree) 

471 with the maximum possible sum of edge weights. A spanning forest is a 

472 union of the spanning trees for each connected component of the graph. 

473 

474 Parameters 

475 ---------- 

476 G : undirected Graph 

477 An undirected graph. If `G` is connected, then the algorithm finds a 

478 spanning tree. Otherwise, a spanning forest is found. 

479 

480 algorithm : string 

481 The algorithm to use when finding a maximum spanning tree. Valid 

482 choices are 'kruskal', 'prim', or 'boruvka'. The default is 'kruskal'. 

483 

484 weight : string 

485 Edge data key to use for weight (default 'weight'). 

486 

487 keys : bool 

488 Whether to yield edge key in multigraphs in addition to the edge. 

489 If `G` is not a multigraph, this is ignored. 

490 

491 data : bool, optional 

492 If True yield the edge data along with the edge. 

493 

494 ignore_nan : bool (default: False) 

495 If a NaN is found as an edge weight normally an exception is raised. 

496 If `ignore_nan is True` then that edge is ignored instead. 

497 

498 Returns 

499 ------- 

500 edges : iterator 

501 An iterator over edges in a maximum spanning tree of `G`. 

502 Edges connecting nodes `u` and `v` are represented as tuples: 

503 `(u, v, k, d)` or `(u, v, k)` or `(u, v, d)` or `(u, v)` 

504 

505 If `G` is a multigraph, `keys` indicates whether the edge key `k` will 

506 be reported in the third position in the edge tuple. `data` indicates 

507 whether the edge datadict `d` will appear at the end of the edge tuple. 

508 

509 If `G` is not a multigraph, the tuples are `(u, v, d)` if `data` is True 

510 or `(u, v)` if `data` is False. 

511 

512 Examples 

513 -------- 

514 >>> from networkx.algorithms import tree 

515 

516 Find maximum spanning edges by Kruskal's algorithm 

517 

518 >>> G = nx.cycle_graph(4) 

519 >>> G.add_edge(0, 3, weight=2) 

520 >>> mst = tree.maximum_spanning_edges(G, algorithm="kruskal", data=False) 

521 >>> edgelist = list(mst) 

522 >>> sorted(sorted(e) for e in edgelist) 

523 [[0, 1], [0, 3], [1, 2]] 

524 

525 Find maximum spanning edges by Prim's algorithm 

526 

527 >>> G = nx.cycle_graph(4) 

528 >>> G.add_edge(0, 3, weight=2) # assign weight 2 to edge 0-3 

529 >>> mst = tree.maximum_spanning_edges(G, algorithm="prim", data=False) 

530 >>> edgelist = list(mst) 

531 >>> sorted(sorted(e) for e in edgelist) 

532 [[0, 1], [0, 3], [2, 3]] 

533 

534 Notes 

535 ----- 

536 For Borůvka's algorithm, each edge must have a weight attribute, and 

537 each edge weight must be distinct. 

538 

539 For the other algorithms, if the graph edges do not have a weight 

540 attribute a default weight of 1 will be used. 

541 

542 Modified code from David Eppstein, April 2006 

543 http://www.ics.uci.edu/~eppstein/PADS/ 

544 """ 

545 try: 

546 algo = ALGORITHMS[algorithm] 

547 except KeyError as err: 

548 msg = f"{algorithm} is not a valid choice for an algorithm." 

549 raise ValueError(msg) from err 

550 

551 return algo( 

552 G, minimum=False, weight=weight, keys=keys, data=data, ignore_nan=ignore_nan 

553 ) 

554 

555 

556@nx._dispatchable(preserve_all_attrs=True, returns_graph=True) 

557def minimum_spanning_tree(G, weight="weight", algorithm="kruskal", ignore_nan=False): 

558 """Returns a minimum spanning tree or forest on an undirected graph `G`. 

559 

560 Parameters 

561 ---------- 

562 G : undirected graph 

563 An undirected graph. If `G` is connected, then the algorithm finds a 

564 spanning tree. Otherwise, a spanning forest is found. 

565 

566 weight : str 

567 Data key to use for edge weights. 

568 

569 algorithm : string 

570 The algorithm to use when finding a minimum spanning tree. Valid 

571 choices are 'kruskal', 'prim', or 'boruvka'. The default is 

572 'kruskal'. 

573 

574 ignore_nan : bool (default: False) 

575 If a NaN is found as an edge weight normally an exception is raised. 

576 If `ignore_nan is True` then that edge is ignored instead. 

577 

578 Returns 

579 ------- 

580 G : NetworkX Graph 

581 A minimum spanning tree or forest. 

582 

583 Examples 

584 -------- 

585 >>> G = nx.cycle_graph(4) 

586 >>> G.add_edge(0, 3, weight=2) 

587 >>> T = nx.minimum_spanning_tree(G) 

588 >>> sorted(T.edges(data=True)) 

589 [(0, 1, {}), (1, 2, {}), (2, 3, {})] 

590 

591 

592 Notes 

593 ----- 

594 For Borůvka's algorithm, each edge must have a weight attribute, and 

595 each edge weight must be distinct. 

596 

597 For the other algorithms, if the graph edges do not have a weight 

598 attribute a default weight of 1 will be used. 

599 

600 There may be more than one tree with the same minimum or maximum weight. 

601 See :mod:`networkx.tree.recognition` for more detailed definitions. 

602 

603 Isolated nodes with self-loops are in the tree as edgeless isolated nodes. 

604 

605 """ 

606 edges = minimum_spanning_edges( 

607 G, algorithm, weight, keys=True, data=True, ignore_nan=ignore_nan 

608 ) 

609 T = G.__class__() # Same graph class as G 

610 T.graph.update(G.graph) 

611 T.add_nodes_from(G.nodes.items()) 

612 T.add_edges_from(edges) 

613 return T 

614 

615 

616@nx._dispatchable(preserve_all_attrs=True, returns_graph=True) 

617def partition_spanning_tree( 

618 G, minimum=True, weight="weight", partition="partition", ignore_nan=False 

619): 

620 """ 

621 Find a spanning tree while respecting a partition of edges. 

622 

623 Edges can be flagged as either `INCLUDED` which are required to be in the 

624 returned tree, `EXCLUDED`, which cannot be in the returned tree and `OPEN`. 

625 

626 This is used in the SpanningTreeIterator to create new partitions following 

627 the algorithm of Sörensen and Janssens [1]_. 

628 

629 Parameters 

630 ---------- 

631 G : undirected graph 

632 An undirected graph. 

633 

634 minimum : bool (default: True) 

635 Determines whether the returned tree is the minimum spanning tree of 

636 the partition of the maximum one. 

637 

638 weight : str 

639 Data key to use for edge weights. 

640 

641 partition : str 

642 The key for the edge attribute containing the partition 

643 data on the graph. Edges can be included, excluded or open using the 

644 `EdgePartition` enum. 

645 

646 ignore_nan : bool (default: False) 

647 If a NaN is found as an edge weight normally an exception is raised. 

648 If `ignore_nan is True` then that edge is ignored instead. 

649 

650 

651 Returns 

652 ------- 

653 G : NetworkX Graph 

654 A minimum spanning tree using all of the included edges in the graph and 

655 none of the excluded edges. 

656 

657 References 

658 ---------- 

659 .. [1] G.K. Janssens, K. Sörensen, An algorithm to generate all spanning 

660 trees in order of increasing cost, Pesquisa Operacional, 2005-08, 

661 Vol. 25 (2), p. 219-229, 

662 https://www.scielo.br/j/pope/a/XHswBwRwJyrfL88dmMwYNWp/?lang=en 

663 """ 

664 edges = kruskal_mst_edges( 

665 G, 

666 minimum, 

667 weight, 

668 keys=True, 

669 data=True, 

670 ignore_nan=ignore_nan, 

671 partition=partition, 

672 ) 

673 T = G.__class__() # Same graph class as G 

674 T.graph.update(G.graph) 

675 T.add_nodes_from(G.nodes.items()) 

676 T.add_edges_from(edges) 

677 return T 

678 

679 

680@nx._dispatchable(preserve_all_attrs=True, returns_graph=True) 

681def maximum_spanning_tree(G, weight="weight", algorithm="kruskal", ignore_nan=False): 

682 """Returns a maximum spanning tree or forest on an undirected graph `G`. 

683 

684 Parameters 

685 ---------- 

686 G : undirected graph 

687 An undirected graph. If `G` is connected, then the algorithm finds a 

688 spanning tree. Otherwise, a spanning forest is found. 

689 

690 weight : str 

691 Data key to use for edge weights. 

692 

693 algorithm : string 

694 The algorithm to use when finding a maximum spanning tree. Valid 

695 choices are 'kruskal', 'prim', or 'boruvka'. The default is 

696 'kruskal'. 

697 

698 ignore_nan : bool (default: False) 

699 If a NaN is found as an edge weight normally an exception is raised. 

700 If `ignore_nan is True` then that edge is ignored instead. 

701 

702 

703 Returns 

704 ------- 

705 G : NetworkX Graph 

706 A maximum spanning tree or forest. 

707 

708 

709 Examples 

710 -------- 

711 >>> G = nx.cycle_graph(4) 

712 >>> G.add_edge(0, 3, weight=2) 

713 >>> T = nx.maximum_spanning_tree(G) 

714 >>> sorted(T.edges(data=True)) 

715 [(0, 1, {}), (0, 3, {'weight': 2}), (1, 2, {})] 

716 

717 

718 Notes 

719 ----- 

720 For Borůvka's algorithm, each edge must have a weight attribute, and 

721 each edge weight must be distinct. 

722 

723 For the other algorithms, if the graph edges do not have a weight 

724 attribute a default weight of 1 will be used. 

725 

726 There may be more than one tree with the same minimum or maximum weight. 

727 See :mod:`networkx.tree.recognition` for more detailed definitions. 

728 

729 Isolated nodes with self-loops are in the tree as edgeless isolated nodes. 

730 

731 """ 

732 edges = maximum_spanning_edges( 

733 G, algorithm, weight, keys=True, data=True, ignore_nan=ignore_nan 

734 ) 

735 edges = list(edges) 

736 T = G.__class__() # Same graph class as G 

737 T.graph.update(G.graph) 

738 T.add_nodes_from(G.nodes.items()) 

739 T.add_edges_from(edges) 

740 return T 

741 

742 

743@py_random_state(3) 

744@nx._dispatchable(preserve_edge_attrs=True, returns_graph=True) 

745def random_spanning_tree(G, weight=None, *, multiplicative=True, seed=None): 

746 """ 

747 Sample a random spanning tree using the edges weights of `G`. 

748 

749 This function supports two different methods for determining the 

750 probability of the graph. If ``multiplicative=True``, the probability 

751 is based on the product of edge weights, and if ``multiplicative=False`` 

752 it is based on the sum of the edge weight. However, since it is 

753 easier to determine the total weight of all spanning trees for the 

754 multiplicative version, that is significantly faster and should be used if 

755 possible. Additionally, setting `weight` to `None` will cause a spanning tree 

756 to be selected with uniform probability. 

757 

758 The function uses algorithm A8 in [1]_ . 

759 

760 Parameters 

761 ---------- 

762 G : nx.Graph 

763 An undirected version of the original graph. 

764 

765 weight : string 

766 The edge key for the edge attribute holding edge weight. 

767 

768 multiplicative : bool, default=True 

769 If `True`, the probability of each tree is the product of its edge weight 

770 over the sum of the product of all the spanning trees in the graph. If 

771 `False`, the probability is the sum of its edge weight over the sum of 

772 the sum of weights for all spanning trees in the graph. 

773 

774 seed : integer, random_state, or None (default) 

775 Indicator of random number generation state. 

776 See :ref:`Randomness<randomness>`. 

777 

778 Returns 

779 ------- 

780 nx.Graph 

781 A spanning tree using the distribution defined by the weight of the tree. 

782 

783 References 

784 ---------- 

785 .. [1] V. Kulkarni, Generating random combinatorial objects, Journal of 

786 Algorithms, 11 (1990), pp. 185–207 

787 """ 

788 

789 def find_node(merged_nodes, node): 

790 """ 

791 We can think of clusters of contracted nodes as having one 

792 representative in the graph. Each node which is not in merged_nodes 

793 is still its own representative. Since a representative can be later 

794 contracted, we need to recursively search though the dict to find 

795 the final representative, but once we know it we can use path 

796 compression to speed up the access of the representative for next time. 

797 

798 This cannot be replaced by the standard NetworkX union_find since that 

799 data structure will merge nodes with less representing nodes into the 

800 one with more representing nodes but this function requires we merge 

801 them using the order that contract_edges contracts using. 

802 

803 Parameters 

804 ---------- 

805 merged_nodes : dict 

806 The dict storing the mapping from node to representative 

807 node 

808 The node whose representative we seek 

809 

810 Returns 

811 ------- 

812 The representative of the `node` 

813 """ 

814 if node not in merged_nodes: 

815 return node 

816 else: 

817 rep = find_node(merged_nodes, merged_nodes[node]) 

818 merged_nodes[node] = rep 

819 return rep 

820 

821 def prepare_graph(): 

822 """ 

823 For the graph `G`, remove all edges not in the set `V` and then 

824 contract all edges in the set `U`. 

825 

826 Returns 

827 ------- 

828 A copy of `G` which has had all edges not in `V` removed and all edges 

829 in `U` contracted. 

830 """ 

831 

832 # The result is a MultiGraph version of G so that parallel edges are 

833 # allowed during edge contraction 

834 result = nx.MultiGraph(incoming_graph_data=G) 

835 

836 # Remove all edges not in V 

837 edges_to_remove = set(result.edges()).difference(V) 

838 result.remove_edges_from(edges_to_remove) 

839 

840 # Contract all edges in U 

841 # 

842 # Imagine that you have two edges to contract and they share an 

843 # endpoint like this: 

844 # [0] ----- [1] ----- [2] 

845 # If we contract (0, 1) first, the contraction function will always 

846 # delete the second node it is passed so the resulting graph would be 

847 # [0] ----- [2] 

848 # and edge (1, 2) no longer exists but (0, 2) would need to be contracted 

849 # in its place now. That is why I use the below dict as a merge-find 

850 # data structure with path compression to track how the nodes are merged. 

851 merged_nodes = {} 

852 

853 for u, v in U: 

854 u_rep = find_node(merged_nodes, u) 

855 v_rep = find_node(merged_nodes, v) 

856 # We cannot contract a node with itself 

857 if u_rep == v_rep: 

858 continue 

859 nx.contracted_nodes(result, u_rep, v_rep, self_loops=False, copy=False) 

860 merged_nodes[v_rep] = u_rep 

861 

862 return merged_nodes, result 

863 

864 def spanning_tree_total_weight(G, weight): 

865 """ 

866 Find the sum of weights of the spanning trees of `G` using the 

867 appropriate `method`. 

868 

869 This is easy if the chosen method is 'multiplicative', since we can 

870 use Kirchhoff's Tree Matrix Theorem directly. However, with the 

871 'additive' method, this process is slightly more complex and less 

872 computationally efficient as we have to find the number of spanning 

873 trees which contain each possible edge in the graph. 

874 

875 Parameters 

876 ---------- 

877 G : NetworkX Graph 

878 The graph to find the total weight of all spanning trees on. 

879 

880 weight : string 

881 The key for the weight edge attribute of the graph. 

882 

883 Returns 

884 ------- 

885 float 

886 The sum of either the multiplicative or additive weight for all 

887 spanning trees in the graph. 

888 """ 

889 if multiplicative: 

890 return number_of_spanning_trees(G, weight=weight) 

891 else: 

892 # There are two cases for the total spanning tree additive weight. 

893 # 1. There is one edge in the graph. Then the only spanning tree is 

894 # that edge itself, which will have a total weight of that edge 

895 # itself. 

896 if G.number_of_edges() == 1: 

897 return G.edges(data=weight).__iter__().__next__()[2] 

898 # 2. There are no edges or two or more edges in the graph. Then, we find the 

899 # total weight of the spanning trees using the formula in the 

900 # reference paper: take the weight of each edge and multiply it by 

901 # the number of spanning trees which include that edge. This 

902 # can be accomplished by contracting the edge and finding the 

903 # multiplicative total spanning tree weight if the weight of each edge 

904 # is assumed to be 1, which is conveniently built into networkx already, 

905 # by calling number_of_spanning_trees with weight=None. 

906 # Note that with no edges the returned value is just zero. 

907 else: 

908 total = 0 

909 for u, v, w in G.edges(data=weight): 

910 total += w * nx.number_of_spanning_trees( 

911 nx.contracted_edge(G, edge=(u, v), self_loops=False), 

912 weight=None, 

913 ) 

914 return total 

915 

916 if G.number_of_nodes() < 2: 

917 # no edges in the spanning tree 

918 return nx.empty_graph(G.nodes) 

919 

920 U = set() 

921 st_cached_value = 0 

922 V = set(G.edges()) 

923 shuffled_edges = list(G.edges()) 

924 seed.shuffle(shuffled_edges) 

925 

926 for u, v in shuffled_edges: 

927 e_weight = G[u][v][weight] if weight is not None else 1 

928 node_map, prepared_G = prepare_graph() 

929 G_total_tree_weight = spanning_tree_total_weight(prepared_G, weight) 

930 # Add the edge to U so that we can compute the total tree weight 

931 # assuming we include that edge 

932 # Now, if (u, v) cannot exist in G because it is fully contracted out 

933 # of existence, then it by definition cannot influence G_e's Kirchhoff 

934 # value. But, we also cannot pick it. 

935 rep_edge = (find_node(node_map, u), find_node(node_map, v)) 

936 # Check to see if the 'representative edge' for the current edge is 

937 # in prepared_G. If so, then we can pick it. 

938 if rep_edge in prepared_G.edges: 

939 prepared_G_e = nx.contracted_edge( 

940 prepared_G, edge=rep_edge, self_loops=False 

941 ) 

942 G_e_total_tree_weight = spanning_tree_total_weight(prepared_G_e, weight) 

943 if multiplicative: 

944 threshold = e_weight * G_e_total_tree_weight / G_total_tree_weight 

945 else: 

946 numerator = (st_cached_value + e_weight) * nx.number_of_spanning_trees( 

947 prepared_G_e 

948 ) + G_e_total_tree_weight 

949 denominator = ( 

950 st_cached_value * nx.number_of_spanning_trees(prepared_G) 

951 + G_total_tree_weight 

952 ) 

953 threshold = numerator / denominator 

954 else: 

955 threshold = 0.0 

956 z = seed.uniform(0.0, 1.0) 

957 if z > threshold: 

958 # Remove the edge from V since we did not pick it. 

959 V.remove((u, v)) 

960 else: 

961 # Add the edge to U since we picked it. 

962 st_cached_value += e_weight 

963 U.add((u, v)) 

964 # If we decide to keep an edge, it may complete the spanning tree. 

965 if len(U) == G.number_of_nodes() - 1: 

966 spanning_tree = nx.Graph() 

967 spanning_tree.add_edges_from(U) 

968 return spanning_tree 

969 raise Exception(f"Something went wrong! Only {len(U)} edges in the spanning tree!") 

970 

971 

972class SpanningTreeIterator: 

973 """ 

974 Iterate over all spanning trees of a graph in either increasing or 

975 decreasing cost. 

976 

977 Notes 

978 ----- 

979 This iterator uses the partition scheme from [1]_ (included edges, 

980 excluded edges and open edges) as well as a modified Kruskal's Algorithm 

981 to generate minimum spanning trees which respect the partition of edges. 

982 For spanning trees with the same weight, ties are broken arbitrarily. 

983 

984 References 

985 ---------- 

986 .. [1] G.K. Janssens, K. Sörensen, An algorithm to generate all spanning 

987 trees in order of increasing cost, Pesquisa Operacional, 2005-08, 

988 Vol. 25 (2), p. 219-229, 

989 https://www.scielo.br/j/pope/a/XHswBwRwJyrfL88dmMwYNWp/?lang=en 

990 """ 

991 

992 @dataclass(order=True) 

993 class Partition: 

994 """ 

995 This dataclass represents a partition and stores a dict with the edge 

996 data and the weight of the minimum spanning tree of the partition dict. 

997 """ 

998 

999 mst_weight: float 

1000 partition_dict: dict = field(compare=False) 

1001 

1002 def __copy__(self): 

1003 return SpanningTreeIterator.Partition( 

1004 self.mst_weight, self.partition_dict.copy() 

1005 ) 

1006 

1007 def __init__(self, G, weight="weight", minimum=True, ignore_nan=False): 

1008 """ 

1009 Initialize the iterator 

1010 

1011 Parameters 

1012 ---------- 

1013 G : nx.Graph 

1014 The directed graph which we need to iterate trees over 

1015 

1016 weight : String, default = "weight" 

1017 The edge attribute used to store the weight of the edge 

1018 

1019 minimum : bool, default = True 

1020 Return the trees in increasing order while true and decreasing order 

1021 while false. 

1022 

1023 ignore_nan : bool, default = False 

1024 If a NaN is found as an edge weight normally an exception is raised. 

1025 If `ignore_nan is True` then that edge is ignored instead. 

1026 """ 

1027 self.G = G.copy() 

1028 self.G.__networkx_cache__ = None # Disable caching 

1029 self.weight = weight 

1030 self.minimum = minimum 

1031 self.ignore_nan = ignore_nan 

1032 # Randomly create a key for an edge attribute to hold the partition data 

1033 self.partition_key = ( 

1034 "SpanningTreeIterators super secret partition attribute name" 

1035 )