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 """ 

196 Sort the edges of the graph with respect to the partition data. 

197 Edges are returned in the following order: 

198 

199 * Included edges 

200 * Open edges from smallest to largest weight 

201 * Excluded edges 

202 """ 

203 included_edges = [] 

204 open_edges = [] 

205 for e in edges: 

206 d = e[-1] 

207 wt = d.get(weight, 1) 

208 if isnan(wt): 

209 if ignore_nan: 

210 continue 

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

212 

213 edge = (wt,) + e 

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

215 included_edges.append(edge) 

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

217 continue 

218 else: 

219 open_edges.append(edge) 

220 

221 if minimum: 

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

223 else: 

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

225 

226 # Condense the lists into one 

227 included_edges.extend(sorted_open_edges) 

228 sorted_edges = included_edges 

229 del open_edges, sorted_open_edges, included_edges 

230 

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

232 if G.is_multigraph(): 

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

234 if subtrees[u] != subtrees[v]: 

235 if keys: 

236 if data: 

237 yield u, v, k, d 

238 else: 

239 yield u, v, k 

240 else: 

241 if data: 

242 yield u, v, d 

243 else: 

244 yield u, v 

245 subtrees.union(u, v) 

246 else: 

247 for wt, u, v, d in sorted_edges: 

248 if subtrees[u] != subtrees[v]: 

249 if data: 

250 yield u, v, d 

251 else: 

252 yield u, v 

253 subtrees.union(u, v) 

254 

255 

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

257def prim_mst_edges(G, minimum, weight="weight", keys=True, data=True, ignore_nan=False): 

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

259 

260 Parameters 

261 ---------- 

262 G : NetworkX Graph 

263 The graph holding the tree of interest. 

264 

265 minimum : bool (default: True) 

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

267 

268 weight : string (default: 'weight') 

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

270 

271 keys : bool (default: True) 

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

273 Otherwise `keys` is ignored. 

274 

275 data : bool (default: True) 

276 Flag for whether to yield edge attribute dicts. 

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

278 If False, yield edges `(u, v)`. 

279 

280 ignore_nan : bool (default: False) 

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

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

283 

284 """ 

285 is_multigraph = G.is_multigraph() 

286 

287 nodes = set(G) 

288 c = count() 

289 

290 sign = 1 if minimum else -1 

291 

292 while nodes: 

293 u = nodes.pop() 

294 frontier = [] 

295 visited = {u} 

296 if is_multigraph: 

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

298 for k, d in keydict.items(): 

299 wt = d.get(weight, 1) * sign 

300 if isnan(wt): 

301 if ignore_nan: 

302 continue 

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

304 raise ValueError(msg) 

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

306 else: 

307 for v, d in G.adj[u].items(): 

308 wt = d.get(weight, 1) * sign 

309 if isnan(wt): 

310 if ignore_nan: 

311 continue 

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

313 raise ValueError(msg) 

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

315 while nodes and frontier: 

316 if is_multigraph: 

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

318 else: 

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

320 if v in visited or v not in nodes: 

321 continue 

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

323 if is_multigraph and keys: 

324 if data: 

325 yield u, v, k, d 

326 else: 

327 yield u, v, k 

328 else: 

329 if data: 

330 yield u, v, d 

331 else: 

332 yield u, v 

333 # update frontier 

334 visited.add(v) 

335 nodes.discard(v) 

336 if is_multigraph: 

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

338 if w in visited: 

339 continue 

340 for k2, d2 in keydict.items(): 

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

342 if isnan(new_weight): 

343 if ignore_nan: 

344 continue 

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

346 raise ValueError(msg) 

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

348 else: 

349 for w, d2 in G.adj[v].items(): 

350 if w in visited: 

351 continue 

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

353 if isnan(new_weight): 

354 if ignore_nan: 

355 continue 

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

357 raise ValueError(msg) 

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

359 

360 

361ALGORITHMS = { 

362 "boruvka": boruvka_mst_edges, 

363 "borůvka": boruvka_mst_edges, 

364 "kruskal": kruskal_mst_edges, 

365 "prim": prim_mst_edges, 

366} 

367 

368 

369@not_implemented_for("directed") 

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

371def minimum_spanning_edges( 

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

373): 

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

375 weighted graph. 

376 

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

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

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

380 

381 Parameters 

382 ---------- 

383 G : undirected Graph 

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

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

386 

387 algorithm : string 

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

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

390 

391 weight : string 

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

393 

394 keys : bool 

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

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

397 

398 data : bool, optional 

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

400 

401 ignore_nan : bool (default: False) 

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

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

404 

405 Returns 

406 ------- 

407 edges : iterator 

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

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

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

411 

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

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

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

415 

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

417 or `(u, v)` if `data` is False. 

418 

419 Examples 

420 -------- 

421 >>> from networkx.algorithms import tree 

422 

423 Find minimum spanning edges by Kruskal's algorithm 

424 

425 >>> G = nx.cycle_graph(4) 

426 >>> G.add_edge(0, 3, weight=2) 

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

428 >>> edgelist = list(mst) 

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

430 [[0, 1], [1, 2], [2, 3]] 

431 

432 Find minimum spanning edges by Prim's algorithm 

433 

434 >>> G = nx.cycle_graph(4) 

435 >>> G.add_edge(0, 3, weight=2) 

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

437 >>> edgelist = list(mst) 

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

439 [[0, 1], [1, 2], [2, 3]] 

440 

441 Notes 

442 ----- 

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

444 each edge weight must be distinct. 

445 

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

447 attribute a default weight of 1 will be used. 

448 

449 Modified code from David Eppstein, April 2006 

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

451 

452 """ 

453 try: 

454 algo = ALGORITHMS[algorithm] 

455 except KeyError as err: 

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

457 raise ValueError(msg) from err 

458 

459 return algo( 

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

461 ) 

462 

463 

464@not_implemented_for("directed") 

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

466def maximum_spanning_edges( 

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

468): 

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

470 weighted graph. 

471 

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

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

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

475 

476 Parameters 

477 ---------- 

478 G : undirected Graph 

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

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

481 

482 algorithm : string 

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

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

485 

486 weight : string 

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

488 

489 keys : bool 

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

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

492 

493 data : bool, optional 

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

495 

496 ignore_nan : bool (default: False) 

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

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

499 

500 Returns 

501 ------- 

502 edges : iterator 

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

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

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

506 

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

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

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

510 

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

512 or `(u, v)` if `data` is False. 

513 

514 Examples 

515 -------- 

516 >>> from networkx.algorithms import tree 

517 

518 Find maximum spanning edges by Kruskal's algorithm 

519 

520 >>> G = nx.cycle_graph(4) 

521 >>> G.add_edge(0, 3, weight=2) 

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

523 >>> edgelist = list(mst) 

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

525 [[0, 1], [0, 3], [1, 2]] 

526 

527 Find maximum spanning edges by Prim's algorithm 

528 

529 >>> G = nx.cycle_graph(4) 

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

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

532 >>> edgelist = list(mst) 

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

534 [[0, 1], [0, 3], [2, 3]] 

535 

536 Notes 

537 ----- 

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

539 each edge weight must be distinct. 

540 

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

542 attribute a default weight of 1 will be used. 

543 

544 Modified code from David Eppstein, April 2006 

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

546 """ 

547 try: 

548 algo = ALGORITHMS[algorithm] 

549 except KeyError as err: 

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

551 raise ValueError(msg) from err 

552 

553 return algo( 

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

555 ) 

556 

557 

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

559def minimum_spanning_tree(G, weight="weight", algorithm="kruskal", ignore_nan=False): 

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

561 

562 Parameters 

563 ---------- 

564 G : undirected graph 

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

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

567 

568 weight : str 

569 Data key to use for edge weights. 

570 

571 algorithm : string 

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

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

574 'kruskal'. 

575 

576 ignore_nan : bool (default: False) 

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

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

579 

580 Returns 

581 ------- 

582 G : NetworkX Graph 

583 A minimum spanning tree or forest. 

584 

585 Examples 

586 -------- 

587 >>> G = nx.cycle_graph(4) 

588 >>> G.add_edge(0, 3, weight=2) 

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

590 >>> sorted(T.edges(data=True)) 

591 [(0, 1, {}), (1, 2, {}), (2, 3, {})] 

592 

593 

594 Notes 

595 ----- 

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

597 each edge weight must be distinct. 

598 

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

600 attribute a default weight of 1 will be used. 

601 

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

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

604 

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

606 

607 """ 

608 edges = minimum_spanning_edges( 

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

610 ) 

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

612 T.graph.update(G.graph) 

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

614 T.add_edges_from(edges) 

615 return T 

616 

617 

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

619def partition_spanning_tree( 

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

621): 

622 """ 

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

624 

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

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

627 

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

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

630 

631 Parameters 

632 ---------- 

633 G : undirected graph 

634 An undirected graph. 

635 

636 minimum : bool (default: True) 

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

638 the partition of the maximum one. 

639 

640 weight : str 

641 Data key to use for edge weights. 

642 

643 partition : str 

644 The key for the edge attribute containing the partition 

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

646 `EdgePartition` enum. 

647 

648 ignore_nan : bool (default: False) 

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

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

651 

652 

653 Returns 

654 ------- 

655 G : NetworkX Graph 

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

657 none of the excluded edges. 

658 

659 References 

660 ---------- 

661 .. [1] G.K. Janssens, K. Sörensen, An algorithm to generate all spanning 

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

663 Vol. 25 (2), p. 219-229, 

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

665 """ 

666 edges = kruskal_mst_edges( 

667 G, 

668 minimum, 

669 weight, 

670 keys=True, 

671 data=True, 

672 ignore_nan=ignore_nan, 

673 partition=partition, 

674 ) 

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

676 T.graph.update(G.graph) 

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

678 T.add_edges_from(edges) 

679 return T 

680 

681 

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

683def maximum_spanning_tree(G, weight="weight", algorithm="kruskal", ignore_nan=False): 

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

685 

686 Parameters 

687 ---------- 

688 G : undirected graph 

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

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

691 

692 weight : str 

693 Data key to use for edge weights. 

694 

695 algorithm : string 

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

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

698 'kruskal'. 

699 

700 ignore_nan : bool (default: False) 

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

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

703 

704 

705 Returns 

706 ------- 

707 G : NetworkX Graph 

708 A maximum spanning tree or forest. 

709 

710 

711 Examples 

712 -------- 

713 >>> G = nx.cycle_graph(4) 

714 >>> G.add_edge(0, 3, weight=2) 

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

716 >>> sorted(T.edges(data=True)) 

717 [(0, 1, {}), (0, 3, {'weight': 2}), (1, 2, {})] 

718 

719 

720 Notes 

721 ----- 

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

723 each edge weight must be distinct. 

724 

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

726 attribute a default weight of 1 will be used. 

727 

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

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

730 

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

732 

733 """ 

734 edges = maximum_spanning_edges( 

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

736 ) 

737 edges = list(edges) 

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

739 T.graph.update(G.graph) 

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

741 T.add_edges_from(edges) 

742 return T 

743 

744 

745@py_random_state(3) 

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

747def random_spanning_tree(G, weight=None, *, multiplicative=True, seed=None): 

748 """ 

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

750 

751 This function supports two different methods for determining the 

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

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

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

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

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

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

758 to be selected with uniform probability. 

759 

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

761 

762 Parameters 

763 ---------- 

764 G : nx.Graph 

765 An undirected version of the original graph. 

766 

767 weight : string 

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

769 

770 multiplicative : bool, default=True 

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

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

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

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

775 

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

777 Indicator of random number generation state. 

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

779 

780 Returns 

781 ------- 

782 nx.Graph 

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

784 

785 References 

786 ---------- 

787 .. [1] V. Kulkarni, Generating random combinatorial objects, Journal of 

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

789 """ 

790 

791 def find_node(merged_nodes, node): 

792 """ 

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

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

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

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

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

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

799 

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

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

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

803 them using the order that contract_edges contracts using. 

804 

805 Parameters 

806 ---------- 

807 merged_nodes : dict 

808 The dict storing the mapping from node to representative 

809 node 

810 The node whose representative we seek 

811 

812 Returns 

813 ------- 

814 The representative of the `node` 

815 """ 

816 if node not in merged_nodes: 

817 return node 

818 else: 

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

820 merged_nodes[node] = rep 

821 return rep 

822 

823 def prepare_graph(): 

824 """ 

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

826 contract all edges in the set `U`. 

827 

828 Returns 

829 ------- 

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

831 in `U` contracted. 

832 """ 

833 

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

835 # allowed during edge contraction 

836 result = nx.MultiGraph(incoming_graph_data=G) 

837 

838 # Remove all edges not in V 

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

840 result.remove_edges_from(edges_to_remove) 

841 

842 # Contract all edges in U 

843 # 

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

845 # endpoint like this: 

846 # [0] ----- [1] ----- [2] 

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

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

849 # [0] ----- [2] 

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

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

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

853 merged_nodes = {} 

854 

855 for u, v in U: 

856 u_rep = find_node(merged_nodes, u) 

857 v_rep = find_node(merged_nodes, v) 

858 # We cannot contract a node with itself 

859 if u_rep == v_rep: 

860 continue 

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

862 merged_nodes[v_rep] = u_rep 

863 

864 return merged_nodes, result 

865 

866 def spanning_tree_total_weight(G, weight): 

867 """ 

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

869 appropriate `method`. 

870 

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

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

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

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

875 trees which contain each possible edge in the graph. 

876 

877 Parameters 

878 ---------- 

879 G : NetworkX Graph 

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

881 

882 weight : string 

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

884 

885 Returns 

886 ------- 

887 float 

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

889 spanning trees in the graph. 

890 """ 

891 if multiplicative: 

892 return number_of_spanning_trees(G, weight=weight) 

893 else: 

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

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

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

897 # itself. 

898 if G.number_of_edges() == 1: 

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

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

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

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

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

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

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

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

907 # by calling number_of_spanning_trees with weight=None. 

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

909 else: 

910 total = 0 

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

912 total += w * nx.number_of_spanning_trees( 

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

914 weight=None, 

915 ) 

916 return total 

917 

918 if G.number_of_nodes() < 2: 

919 # no edges in the spanning tree 

920 return nx.empty_graph(G.nodes) 

921 

922 U = set() 

923 st_cached_value = 0 

924 V = set(G.edges()) 

925 shuffled_edges = list(G.edges()) 

926 seed.shuffle(shuffled_edges) 

927 

928 for u, v in shuffled_edges: 

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

930 node_map, prepared_G = prepare_graph() 

931 G_total_tree_weight = spanning_tree_total_weight(prepared_G, weight) 

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

933 # assuming we include that edge 

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

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

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

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

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

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

940 if rep_edge in prepared_G.edges: 

941 prepared_G_e = nx.contracted_edge( 

942 prepared_G, edge=rep_edge, self_loops=False 

943 ) 

944 G_e_total_tree_weight = spanning_tree_total_weight(prepared_G_e, weight) 

945 if multiplicative: 

946 threshold = e_weight * G_e_total_tree_weight / G_total_tree_weight 

947 else: 

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

949 prepared_G_e 

950 ) + G_e_total_tree_weight 

951 denominator = ( 

952 st_cached_value * nx.number_of_spanning_trees(prepared_G) 

953 + G_total_tree_weight 

954 ) 

955 threshold = numerator / denominator 

956 else: 

957 threshold = 0.0 

958 z = seed.uniform(0.0, 1.0) 

959 if z > threshold: 

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

961 V.remove((u, v)) 

962 else: 

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

964 st_cached_value += e_weight 

965 U.add((u, v)) 

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

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

968 spanning_tree = nx.Graph() 

969 spanning_tree.add_edges_from(U) 

970 return spanning_tree 

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

972 

973 

974class SpanningTreeIterator: 

975 """ 

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

977 decreasing cost. 

978 

979 Notes 

980 ----- 

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

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

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

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

985 

986 References 

987 ---------- 

988 .. [1] G.K. Janssens, K. Sörensen, An algorithm to generate all spanning 

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

990 Vol. 25 (2), p. 219-229, 

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

992 """ 

993 

994 @dataclass(order=True) 

995 class Partition: 

996 """ 

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

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

999 """ 

1000 

1001 mst_weight: float 

1002 partition_dict: dict = field(compare=False) 

1003 

1004 def __copy__(self): 

1005 return SpanningTreeIterator.Partition( 

1006 self.mst_weight, self.partition_dict.copy() 

1007 ) 

1008 

1009 def __init__(self, G, weight="weight", minimum=True, ignore_nan=False): 

1010 """ 

1011 Initialize the iterator 

1012 

1013 Parameters 

1014 ---------- 

1015 G : nx.Graph 

1016 The directed graph which we need to iterate trees over 

1017 

1018 weight : String, default = "weight" 

1019 The edge attribute used to store the weight of the edge 

1020 

1021 minimum : bool, default = True 

1022 Return the trees in increasing order while true and decreasing order 

1023 while false. 

1024 

1025 ignore_nan : bool, default = False 

1026 If a NaN is found as an edge weight normally an exception is raised. 

1027 If `ignore_nan is True` then that edge is ignored instead. 

1028 """ 

1029 self.G = G.copy() 

1030 self.G.__networkx_cache__ = None # Disable caching 

1031 self.weight = weight 

1032 self.minimum = minimum 

1033 self.ignore_nan = ignore_nan 

1034 # Randomly create a key for an edge attribute to hold the partition data 

1035 self.partition_key = (