/usr/lib/python3/dist-packages/networkx/algorithms/shortest_paths/weighted.py is in python3-networkx 1.11-1ubuntu2.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 | # -*- coding: utf-8 -*-
"""
Shortest path algorithms for weighed graphs.
"""
__author__ = """\n""".join(['Aric Hagberg <hagberg@lanl.gov>',
'Loïc Séguin-C. <loicseguin@gmail.com>',
'Dan Schult <dschult@colgate.edu>'])
# Copyright (C) 2004-2015 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
__all__ = ['dijkstra_path',
'dijkstra_path_length',
'bidirectional_dijkstra',
'single_source_dijkstra',
'single_source_dijkstra_path',
'single_source_dijkstra_path_length',
'all_pairs_dijkstra_path',
'all_pairs_dijkstra_path_length',
'dijkstra_predecessor_and_distance',
'bellman_ford',
'negative_edge_cycle',
'goldberg_radzik',
'johnson']
from collections import deque
from heapq import heappush, heappop
from itertools import count
import networkx as nx
from networkx.utils import generate_unique_node
def dijkstra_path(G, source, target, weight='weight'):
"""Returns the shortest path from source to target in a weighted graph G.
Parameters
----------
G : NetworkX graph
source : node
Starting node
target : node
Ending node
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
Returns
-------
path : list
List of nodes in a shortest path.
Raises
------
NetworkXNoPath
If no path exists between source and target.
Examples
--------
>>> G=nx.path_graph(5)
>>> print(nx.dijkstra_path(G,0,4))
[0, 1, 2, 3, 4]
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
See Also
--------
bidirectional_dijkstra()
"""
(length, path) = single_source_dijkstra(G, source, target=target,
weight=weight)
try:
return path[target]
except KeyError:
raise nx.NetworkXNoPath(
"node %s not reachable from %s" % (source, target))
def dijkstra_path_length(G, source, target, weight='weight'):
"""Returns the shortest path length from source to target
in a weighted graph.
Parameters
----------
G : NetworkX graph
source : node label
starting node for path
target : node label
ending node for path
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
Returns
-------
length : number
Shortest path length.
Raises
------
NetworkXNoPath
If no path exists between source and target.
Examples
--------
>>> G=nx.path_graph(5)
>>> print(nx.dijkstra_path_length(G,0,4))
4
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
See Also
--------
bidirectional_dijkstra()
"""
length = single_source_dijkstra_path_length(G, source, weight=weight)
try:
return length[target]
except KeyError:
raise nx.NetworkXNoPath(
"node %s not reachable from %s" % (source, target))
def single_source_dijkstra_path(G, source, cutoff=None, weight='weight'):
"""Compute shortest path between source and all other reachable
nodes for a weighted graph.
Parameters
----------
G : NetworkX graph
source : node
Starting node for path.
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
cutoff : integer or float, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns
-------
paths : dictionary
Dictionary of shortest path lengths keyed by target.
Examples
--------
>>> G=nx.path_graph(5)
>>> path=nx.single_source_dijkstra_path(G,0)
>>> path[4]
[0, 1, 2, 3, 4]
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
See Also
--------
single_source_dijkstra()
"""
(length, path) = single_source_dijkstra(
G, source, cutoff=cutoff, weight=weight)
return path
def single_source_dijkstra_path_length(G, source, cutoff=None,
weight='weight'):
"""Compute the shortest path length between source and all other
reachable nodes for a weighted graph.
Parameters
----------
G : NetworkX graph
source : node label
Starting node for path
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight.
cutoff : integer or float, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns
-------
length : dictionary
Dictionary of shortest lengths keyed by target.
Examples
--------
>>> G=nx.path_graph(5)
>>> length=nx.single_source_dijkstra_path_length(G,0)
>>> length[4]
4
>>> print(length)
{0: 0, 1: 1, 2: 2, 3: 3, 4: 4}
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
See Also
--------
single_source_dijkstra()
"""
if G.is_multigraph():
get_weight = lambda u, v, data: min(
eattr.get(weight, 1) for eattr in data.values())
else:
get_weight = lambda u, v, data: data.get(weight, 1)
return _dijkstra(G, source, get_weight, cutoff=cutoff)
def single_source_dijkstra(G, source, target=None, cutoff=None, weight='weight'):
"""Compute shortest paths and lengths in a weighted graph G.
Uses Dijkstra's algorithm for shortest paths.
Parameters
----------
G : NetworkX graph
source : node label
Starting node for path
target : node label, optional
Ending node for path
cutoff : integer or float, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns
-------
distance,path : dictionaries
Returns a tuple of two dictionaries keyed by node.
The first dictionary stores distance from the source.
The second stores the path from the source to that node.
Examples
--------
>>> G=nx.path_graph(5)
>>> length,path=nx.single_source_dijkstra(G,0)
>>> print(length[4])
4
>>> print(length)
{0: 0, 1: 1, 2: 2, 3: 3, 4: 4}
>>> path[4]
[0, 1, 2, 3, 4]
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
Based on the Python cookbook recipe (119466) at
http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/119466
This algorithm is not guaranteed to work if edge weights
are negative or are floating point numbers
(overflows and roundoff errors can cause problems).
See Also
--------
single_source_dijkstra_path()
single_source_dijkstra_path_length()
"""
if source == target:
return ({source: 0}, {source: [source]})
if G.is_multigraph():
get_weight = lambda u, v, data: min(
eattr.get(weight, 1) for eattr in data.values())
else:
get_weight = lambda u, v, data: data.get(weight, 1)
paths = {source: [source]} # dictionary of paths
return _dijkstra(G, source, get_weight, paths=paths, cutoff=cutoff,
target=target)
def _dijkstra(G, source, get_weight, pred=None, paths=None, cutoff=None,
target=None):
"""Implementation of Dijkstra's algorithm
Parameters
----------
G : NetworkX graph
source : node label
Starting node for path
get_weight: function
Function for getting edge weight
pred: list, optional(default=None)
List of predecessors of a node
paths: dict, optional (default=None)
Path from the source to a target node.
target : node label, optional
Ending node for path
cutoff : integer or float, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns
-------
distance,path : dictionaries
Returns a tuple of two dictionaries keyed by node.
The first dictionary stores distance from the source.
The second stores the path from the source to that node.
pred,distance : dictionaries
Returns two dictionaries representing a list of predecessors
of a node and the distance to each node.
distance : dictionary
Dictionary of shortest lengths keyed by target.
"""
G_succ = G.succ if G.is_directed() else G.adj
push = heappush
pop = heappop
dist = {} # dictionary of final distances
seen = {source: 0}
c = count()
fringe = [] # use heapq with (distance,label) tuples
push(fringe, (0, next(c), source))
while fringe:
(d, _, v) = pop(fringe)
if v in dist:
continue # already searched this node.
dist[v] = d
if v == target:
break
for u, e in G_succ[v].items():
cost = get_weight(v, u, e)
if cost is None:
continue
vu_dist = dist[v] + get_weight(v, u, e)
if cutoff is not None:
if vu_dist > cutoff:
continue
if u in dist:
if vu_dist < dist[u]:
raise ValueError('Contradictory paths found:',
'negative weights?')
elif u not in seen or vu_dist < seen[u]:
seen[u] = vu_dist
push(fringe, (vu_dist, next(c), u))
if paths is not None:
paths[u] = paths[v] + [u]
if pred is not None:
pred[u] = [v]
elif vu_dist == seen[u]:
if pred is not None:
pred[u].append(v)
if paths is not None:
return (dist, paths)
if pred is not None:
return (pred, dist)
return dist
def dijkstra_predecessor_and_distance(G, source, cutoff=None, weight='weight'):
"""Compute shortest path length and predecessors on shortest paths
in weighted graphs.
Parameters
----------
G : NetworkX graph
source : node label
Starting node for path
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
cutoff : integer or float, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns
-------
pred,distance : dictionaries
Returns two dictionaries representing a list of predecessors
of a node and the distance to each node.
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
The list of predecessors contains more than one element only when
there are more than one shortest paths to the key node.
"""
if G.is_multigraph():
get_weight = lambda u, v, data: min(
eattr.get(weight, 1) for eattr in data.values())
else:
get_weight = lambda u, v, data: data.get(weight, 1)
pred = {source: []} # dictionary of predecessors
return _dijkstra(G, source, get_weight, pred=pred, cutoff=cutoff)
def all_pairs_dijkstra_path_length(G, cutoff=None, weight='weight'):
""" Compute shortest path lengths between all nodes in a weighted graph.
Parameters
----------
G : NetworkX graph
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
cutoff : integer or float, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns
-------
distance : dictionary
Dictionary, keyed by source and target, of shortest path lengths.
Examples
--------
>>> G=nx.path_graph(5)
>>> length=nx.all_pairs_dijkstra_path_length(G)
>>> print(length[1][4])
3
>>> length[1]
{0: 1, 1: 0, 2: 1, 3: 2, 4: 3}
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
The dictionary returned only has keys for reachable node pairs.
"""
length = single_source_dijkstra_path_length
# TODO This can be trivially parallelized.
return {n: length(G, n, cutoff=cutoff, weight=weight) for n in G}
def all_pairs_dijkstra_path(G, cutoff=None, weight='weight'):
""" Compute shortest paths between all nodes in a weighted graph.
Parameters
----------
G : NetworkX graph
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
cutoff : integer or float, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns
-------
distance : dictionary
Dictionary, keyed by source and target, of shortest paths.
Examples
--------
>>> G=nx.path_graph(5)
>>> path=nx.all_pairs_dijkstra_path(G)
>>> print(path[0][4])
[0, 1, 2, 3, 4]
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
See Also
--------
floyd_warshall()
"""
path = single_source_dijkstra_path
# TODO This can be trivially parallelized.
return {n: path(G, n, cutoff=cutoff, weight=weight) for n in G}
def bellman_ford(G, source, weight='weight'):
"""Compute shortest path lengths and predecessors on shortest paths
in weighted graphs.
The algorithm has a running time of O(mn) where n is the number of
nodes and m is the number of edges. It is slower than Dijkstra but
can handle negative edge weights.
Parameters
----------
G : NetworkX graph
The algorithm works for all types of graphs, including directed
graphs and multigraphs.
source: node label
Starting node for path
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
Returns
-------
pred, dist : dictionaries
Returns two dictionaries keyed by node to predecessor in the
path and to the distance from the source respectively.
Raises
------
NetworkXUnbounded
If the (di)graph contains a negative cost (di)cycle, the
algorithm raises an exception to indicate the presence of the
negative cost (di)cycle. Note: any negative weight edge in an
undirected graph is a negative cost cycle.
Examples
--------
>>> import networkx as nx
>>> G = nx.path_graph(5, create_using = nx.DiGraph())
>>> pred, dist = nx.bellman_ford(G, 0)
>>> sorted(pred.items())
[(0, None), (1, 0), (2, 1), (3, 2), (4, 3)]
>>> sorted(dist.items())
[(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]
>>> from nose.tools import assert_raises
>>> G = nx.cycle_graph(5, create_using = nx.DiGraph())
>>> G[1][2]['weight'] = -7
>>> assert_raises(nx.NetworkXUnbounded, nx.bellman_ford, G, 0)
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
The dictionaries returned only have keys for nodes reachable from
the source.
In the case where the (di)graph is not connected, if a component
not containing the source contains a negative cost (di)cycle, it
will not be detected.
"""
if source not in G:
raise KeyError("Node %s is not found in the graph" % source)
for u, v, attr in G.selfloop_edges(data=True):
if attr.get(weight, 1) < 0:
raise nx.NetworkXUnbounded("Negative cost cycle detected.")
dist = {source: 0}
pred = {source: None}
if len(G) == 1:
return pred, dist
return _bellman_ford_relaxation(G, pred, dist, [source], weight)
def _bellman_ford_relaxation(G, pred, dist, source, weight):
"""Relaxation loop for Bellman–Ford algorithm
Parameters
----------
G : NetworkX graph
pred: dict
Keyed by node to predecessor in the path
dist: dict
Keyed by node to the distance from the source
source: list
List of source nodes
weight: string
Edge data key corresponding to the edge weight
Returns
-------
Returns two dictionaries keyed by node to predecessor in the
path and to the distance from the source respectively.
Raises
------
NetworkXUnbounded
If the (di)graph contains a negative cost (di)cycle, the
algorithm raises an exception to indicate the presence of the
negative cost (di)cycle. Note: any negative weight edge in an
undirected graph is a negative cost cycle
"""
if G.is_multigraph():
def get_weight(edge_dict):
return min(eattr.get(weight, 1) for eattr in edge_dict.values())
else:
def get_weight(edge_dict):
return edge_dict.get(weight, 1)
G_succ = G.succ if G.is_directed() else G.adj
inf = float('inf')
n = len(G)
count = {}
q = deque(source)
in_q = set(source)
while q:
u = q.popleft()
in_q.remove(u)
# Skip relaxations if the predecessor of u is in the queue.
if pred[u] not in in_q:
dist_u = dist[u]
for v, e in G_succ[u].items():
dist_v = dist_u + get_weight(e)
if dist_v < dist.get(v, inf):
if v not in in_q:
q.append(v)
in_q.add(v)
count_v = count.get(v, 0) + 1
if count_v == n:
raise nx.NetworkXUnbounded(
"Negative cost cycle detected.")
count[v] = count_v
dist[v] = dist_v
pred[v] = u
return pred, dist
def goldberg_radzik(G, source, weight='weight'):
"""Compute shortest path lengths and predecessors on shortest paths
in weighted graphs.
The algorithm has a running time of O(mn) where n is the number of
nodes and m is the number of edges. It is slower than Dijkstra but
can handle negative edge weights.
Parameters
----------
G : NetworkX graph
The algorithm works for all types of graphs, including directed
graphs and multigraphs.
source: node label
Starting node for path
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
Returns
-------
pred, dist : dictionaries
Returns two dictionaries keyed by node to predecessor in the
path and to the distance from the source respectively.
Raises
------
NetworkXUnbounded
If the (di)graph contains a negative cost (di)cycle, the
algorithm raises an exception to indicate the presence of the
negative cost (di)cycle. Note: any negative weight edge in an
undirected graph is a negative cost cycle.
Examples
--------
>>> import networkx as nx
>>> G = nx.path_graph(5, create_using = nx.DiGraph())
>>> pred, dist = nx.goldberg_radzik(G, 0)
>>> sorted(pred.items())
[(0, None), (1, 0), (2, 1), (3, 2), (4, 3)]
>>> sorted(dist.items())
[(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]
>>> from nose.tools import assert_raises
>>> G = nx.cycle_graph(5, create_using = nx.DiGraph())
>>> G[1][2]['weight'] = -7
>>> assert_raises(nx.NetworkXUnbounded, nx.goldberg_radzik, G, 0)
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
The dictionaries returned only have keys for nodes reachable from
the source.
In the case where the (di)graph is not connected, if a component
not containing the source contains a negative cost (di)cycle, it
will not be detected.
"""
if source not in G:
raise KeyError("Node %s is not found in the graph" % source)
for u, v, attr in G.selfloop_edges(data=True):
if attr.get(weight, 1) < 0:
raise nx.NetworkXUnbounded("Negative cost cycle detected.")
if len(G) == 1:
return {source: None}, {source: 0}
if G.is_multigraph():
def get_weight(edge_dict):
return min(attr.get(weight, 1) for attr in edge_dict.values())
else:
def get_weight(edge_dict):
return edge_dict.get(weight, 1)
if G.is_directed():
G_succ = G.succ
else:
G_succ = G.adj
inf = float('inf')
d = dict((u, inf) for u in G)
d[source] = 0
pred = {source: None}
def topo_sort(relabeled):
"""Topologically sort nodes relabeled in the previous round and detect
negative cycles.
"""
# List of nodes to scan in this round. Denoted by A in Goldberg and
# Radzik's paper.
to_scan = []
# In the DFS in the loop below, neg_count records for each node the
# number of edges of negative reduced costs on the path from a DFS root
# to the node in the DFS forest. The reduced cost of an edge (u, v) is
# defined as d[u] + weight[u][v] - d[v].
#
# neg_count also doubles as the DFS visit marker array.
neg_count = {}
for u in relabeled:
# Skip visited nodes.
if u in neg_count:
continue
d_u = d[u]
# Skip nodes without out-edges of negative reduced costs.
if all(d_u + get_weight(e) >= d[v] for v, e in G_succ[u].items()):
continue
# Nonrecursive DFS that inserts nodes reachable from u via edges of
# nonpositive reduced costs into to_scan in (reverse) topological
# order.
stack = [(u, iter(G_succ[u].items()))]
in_stack = set([u])
neg_count[u] = 0
while stack:
u, it = stack[-1]
try:
v, e = next(it)
except StopIteration:
to_scan.append(u)
stack.pop()
in_stack.remove(u)
continue
t = d[u] + get_weight(e)
d_v = d[v]
if t <= d_v:
is_neg = t < d_v
d[v] = t
pred[v] = u
if v not in neg_count:
neg_count[v] = neg_count[u] + int(is_neg)
stack.append((v, iter(G_succ[v].items())))
in_stack.add(v)
elif (v in in_stack and
neg_count[u] + int(is_neg) > neg_count[v]):
# (u, v) is a back edge, and the cycle formed by the
# path v to u and (u, v) contains at least one edge of
# negative reduced cost. The cycle must be of negative
# cost.
raise nx.NetworkXUnbounded(
'Negative cost cycle detected.')
to_scan.reverse()
return to_scan
def relax(to_scan):
"""Relax out-edges of relabeled nodes.
"""
relabeled = set()
# Scan nodes in to_scan in topological order and relax incident
# out-edges. Add the relabled nodes to labeled.
for u in to_scan:
d_u = d[u]
for v, e in G_succ[u].items():
w_e = get_weight(e)
if d_u + w_e < d[v]:
d[v] = d_u + w_e
pred[v] = u
relabeled.add(v)
return relabeled
# Set of nodes relabled in the last round of scan operations. Denoted by B
# in Goldberg and Radzik's paper.
relabeled = set([source])
while relabeled:
to_scan = topo_sort(relabeled)
relabeled = relax(to_scan)
d = dict((u, d[u]) for u in pred)
return pred, d
def negative_edge_cycle(G, weight='weight'):
"""Return True if there exists a negative edge cycle anywhere in G.
Parameters
----------
G : NetworkX graph
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
Returns
-------
negative_cycle : bool
True if a negative edge cycle exists, otherwise False.
Examples
--------
>>> import networkx as nx
>>> G = nx.cycle_graph(5, create_using = nx.DiGraph())
>>> print(nx.negative_edge_cycle(G))
False
>>> G[1][2]['weight'] = -7
>>> print(nx.negative_edge_cycle(G))
True
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
This algorithm uses bellman_ford() but finds negative cycles
on any component by first adding a new node connected to
every node, and starting bellman_ford on that node. It then
removes that extra node.
"""
newnode = generate_unique_node()
G.add_edges_from([(newnode, n) for n in G])
try:
bellman_ford(G, newnode, weight)
except nx.NetworkXUnbounded:
return True
finally:
G.remove_node(newnode)
return False
def bidirectional_dijkstra(G, source, target, weight='weight'):
"""Dijkstra's algorithm for shortest paths using bidirectional search.
Parameters
----------
G : NetworkX graph
source : node
Starting node.
target : node
Ending node.
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
Returns
-------
length : number
Shortest path length.
Returns a tuple of two dictionaries keyed by node.
The first dictionary stores distance from the source.
The second stores the path from the source to that node.
Raises
------
NetworkXNoPath
If no path exists between source and target.
Examples
--------
>>> G=nx.path_graph(5)
>>> length,path=nx.bidirectional_dijkstra(G,0,4)
>>> print(length)
4
>>> print(path)
[0, 1, 2, 3, 4]
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
In practice bidirectional Dijkstra is much more than twice as fast as
ordinary Dijkstra.
Ordinary Dijkstra expands nodes in a sphere-like manner from the
source. The radius of this sphere will eventually be the length
of the shortest path. Bidirectional Dijkstra will expand nodes
from both the source and the target, making two spheres of half
this radius. Volume of the first sphere is pi*r*r while the
others are 2*pi*r/2*r/2, making up half the volume.
This algorithm is not guaranteed to work if edge weights
are negative or are floating point numbers
(overflows and roundoff errors can cause problems).
See Also
--------
shortest_path
shortest_path_length
"""
if source == target:
return (0, [source])
push = heappush
pop = heappop
# Init: Forward Backward
dists = [{}, {}] # dictionary of final distances
paths = [{source: [source]}, {target: [target]}] # dictionary of paths
fringe = [[], []] # heap of (distance, node) tuples for
# extracting next node to expand
seen = [{source: 0}, {target: 0}] # dictionary of distances to
# nodes seen
c = count()
# initialize fringe heap
push(fringe[0], (0, next(c), source))
push(fringe[1], (0, next(c), target))
# neighs for extracting correct neighbor information
if G.is_directed():
neighs = [G.successors_iter, G.predecessors_iter]
else:
neighs = [G.neighbors_iter, G.neighbors_iter]
# variables to hold shortest discovered path
#finaldist = 1e30000
finalpath = []
dir = 1
while fringe[0] and fringe[1]:
# choose direction
# dir == 0 is forward direction and dir == 1 is back
dir = 1 - dir
# extract closest to expand
(dist, _, v) = pop(fringe[dir])
if v in dists[dir]:
# Shortest path to v has already been found
continue
# update distance
dists[dir][v] = dist # equal to seen[dir][v]
if v in dists[1 - dir]:
# if we have scanned v in both directions we are done
# we have now discovered the shortest path
return (finaldist, finalpath)
for w in neighs[dir](v):
if(dir == 0): # forward
if G.is_multigraph():
minweight = min((dd.get(weight, 1)
for k, dd in G[v][w].items()))
else:
minweight = G[v][w].get(weight, 1)
vwLength = dists[dir][v] + minweight # G[v][w].get(weight,1)
else: # back, must remember to change v,w->w,v
if G.is_multigraph():
minweight = min((dd.get(weight, 1)
for k, dd in G[w][v].items()))
else:
minweight = G[w][v].get(weight, 1)
vwLength = dists[dir][v] + minweight # G[w][v].get(weight,1)
if w in dists[dir]:
if vwLength < dists[dir][w]:
raise ValueError(
"Contradictory paths found: negative weights?")
elif w not in seen[dir] or vwLength < seen[dir][w]:
# relaxing
seen[dir][w] = vwLength
push(fringe[dir], (vwLength, next(c), w))
paths[dir][w] = paths[dir][v] + [w]
if w in seen[0] and w in seen[1]:
# see if this path is better than than the already
# discovered shortest path
totaldist = seen[0][w] + seen[1][w]
if finalpath == [] or finaldist > totaldist:
finaldist = totaldist
revpath = paths[1][w][:]
revpath.reverse()
finalpath = paths[0][w] + revpath[1:]
raise nx.NetworkXNoPath("No path between %s and %s." % (source, target))
def johnson(G, weight='weight'):
"""Compute shortest paths between all nodes in a weighted graph using
Johnson's algorithm.
Parameters
----------
G : NetworkX graph
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight.
Returns
-------
distance : dictionary
Dictionary, keyed by source and target, of shortest paths.
Raises
------
NetworkXError
If given graph is not weighted.
Examples
--------
>>> import networkx as nx
>>> graph = nx.DiGraph()
>>> graph.add_weighted_edges_from([('0', '3', 3), ('0', '1', -5),
... ('0', '2', 2), ('1', '2', 4), ('2', '3', 1)])
>>> paths = nx.johnson(graph, weight='weight')
>>> paths['0']['2']
['0', '1', '2']
Notes
-----
Johnson's algorithm is suitable even for graphs with negative weights. It
works by using the Bellman–Ford algorithm to compute a transformation of
the input graph that removes all negative weights, allowing Dijkstra's
algorithm to be used on the transformed graph.
It may be faster than Floyd - Warshall algorithm in sparse graphs.
Algorithm complexity: O(V^2 * logV + V * E)
See Also
--------
floyd_warshall_predecessor_and_distance
floyd_warshall_numpy
all_pairs_shortest_path
all_pairs_shortest_path_length
all_pairs_dijkstra_path
bellman_ford
"""
if not nx.is_weighted(G, weight=weight):
raise nx.NetworkXError('Graph is not weighted.')
dist = {v: 0 for v in G}
pred = {v: None for v in G}
# Calculate distance of shortest paths
dist_bellman = _bellman_ford_relaxation(G, pred, dist, G.nodes(),
weight)[1]
if G.is_multigraph():
get_weight = lambda u, v, data: (
min(eattr.get(weight, 1) for eattr in data.values()) +
dist_bellman[u] - dist_bellman[v])
else:
get_weight = lambda u, v, data: (data.get(weight, 1) +
dist_bellman[u] - dist_bellman[v])
all_pairs = {v: _dijkstra(G, v, get_weight, paths={v: [v]})[1] for v in G}
return all_pairs
|