Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/shortest_paths/weighted.py @ 0:4f3585e2f14b draft default tip
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author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |
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1 """ |
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2 Shortest path algorithms for weighed graphs. |
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3 """ |
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4 |
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5 from collections import deque |
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6 from heapq import heappush, heappop |
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7 from itertools import count |
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8 import networkx as nx |
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9 from networkx.utils import generate_unique_node |
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10 from networkx.algorithms.shortest_paths.generic import _build_paths_from_predecessors |
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11 |
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12 |
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13 __all__ = [ |
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14 "dijkstra_path", |
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15 "dijkstra_path_length", |
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16 "bidirectional_dijkstra", |
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17 "single_source_dijkstra", |
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18 "single_source_dijkstra_path", |
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19 "single_source_dijkstra_path_length", |
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20 "multi_source_dijkstra", |
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21 "multi_source_dijkstra_path", |
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22 "multi_source_dijkstra_path_length", |
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23 "all_pairs_dijkstra", |
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24 "all_pairs_dijkstra_path", |
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25 "all_pairs_dijkstra_path_length", |
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26 "dijkstra_predecessor_and_distance", |
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27 "bellman_ford_path", |
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28 "bellman_ford_path_length", |
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29 "single_source_bellman_ford", |
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30 "single_source_bellman_ford_path", |
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31 "single_source_bellman_ford_path_length", |
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32 "all_pairs_bellman_ford_path", |
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33 "all_pairs_bellman_ford_path_length", |
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34 "bellman_ford_predecessor_and_distance", |
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35 "negative_edge_cycle", |
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36 "goldberg_radzik", |
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37 "johnson", |
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38 ] |
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39 |
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40 |
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41 def _weight_function(G, weight): |
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42 """Returns a function that returns the weight of an edge. |
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43 |
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44 The returned function is specifically suitable for input to |
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45 functions :func:`_dijkstra` and :func:`_bellman_ford_relaxation`. |
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46 |
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47 Parameters |
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48 ---------- |
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49 G : NetworkX graph. |
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50 |
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51 weight : string or function |
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52 If it is callable, `weight` itself is returned. If it is a string, |
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53 it is assumed to be the name of the edge attribute that represents |
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54 the weight of an edge. In that case, a function is returned that |
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55 gets the edge weight according to the specified edge attribute. |
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56 |
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57 Returns |
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58 ------- |
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59 function |
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60 This function returns a callable that accepts exactly three inputs: |
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61 a node, an node adjacent to the first one, and the edge attribute |
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62 dictionary for the eedge joining those nodes. That function returns |
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63 a number representing the weight of an edge. |
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64 |
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65 If `G` is a multigraph, and `weight` is not callable, the |
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66 minimum edge weight over all parallel edges is returned. If any edge |
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67 does not have an attribute with key `weight`, it is assumed to |
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68 have weight one. |
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69 |
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70 """ |
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71 if callable(weight): |
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72 return weight |
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73 # If the weight keyword argument is not callable, we assume it is a |
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74 # string representing the edge attribute containing the weight of |
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75 # the edge. |
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76 if G.is_multigraph(): |
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77 return lambda u, v, d: min(attr.get(weight, 1) for attr in d.values()) |
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78 return lambda u, v, data: data.get(weight, 1) |
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79 |
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80 |
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81 def dijkstra_path(G, source, target, weight="weight"): |
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82 """Returns the shortest weighted path from source to target in G. |
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83 |
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84 Uses Dijkstra's Method to compute the shortest weighted path |
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85 between two nodes in a graph. |
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86 |
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87 Parameters |
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88 ---------- |
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89 G : NetworkX graph |
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90 |
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91 source : node |
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92 Starting node |
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93 |
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94 target : node |
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95 Ending node |
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96 |
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97 weight : string or function |
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98 If this is a string, then edge weights will be accessed via the |
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99 edge attribute with this key (that is, the weight of the edge |
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100 joining `u` to `v` will be ``G.edges[u, v][weight]``). If no |
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101 such edge attribute exists, the weight of the edge is assumed to |
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102 be one. |
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103 |
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104 If this is a function, the weight of an edge is the value |
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105 returned by the function. The function must accept exactly three |
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106 positional arguments: the two endpoints of an edge and the |
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107 dictionary of edge attributes for that edge. The function must |
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108 return a number. |
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109 |
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110 Returns |
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111 ------- |
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112 path : list |
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113 List of nodes in a shortest path. |
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114 |
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115 Raises |
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116 ------ |
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117 NodeNotFound |
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118 If `source` is not in `G`. |
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119 |
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120 NetworkXNoPath |
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121 If no path exists between source and target. |
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122 |
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123 Examples |
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124 -------- |
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125 >>> G = nx.path_graph(5) |
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126 >>> print(nx.dijkstra_path(G, 0, 4)) |
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127 [0, 1, 2, 3, 4] |
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128 |
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129 Notes |
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130 ----- |
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131 Edge weight attributes must be numerical. |
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132 Distances are calculated as sums of weighted edges traversed. |
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133 |
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134 The weight function can be used to hide edges by returning None. |
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135 So ``weight = lambda u, v, d: 1 if d['color']=="red" else None`` |
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136 will find the shortest red path. |
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137 |
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138 The weight function can be used to include node weights. |
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139 |
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140 >>> def func(u, v, d): |
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141 ... node_u_wt = G.nodes[u].get("node_weight", 1) |
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142 ... node_v_wt = G.nodes[v].get("node_weight", 1) |
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143 ... edge_wt = d.get("weight", 1) |
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144 ... return node_u_wt / 2 + node_v_wt / 2 + edge_wt |
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145 |
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146 In this example we take the average of start and end node |
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147 weights of an edge and add it to the weight of the edge. |
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148 |
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149 The function :func:`single_source_dijkstra` computes both |
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150 path and length-of-path if you need both, use that. |
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151 |
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152 See Also |
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153 -------- |
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154 bidirectional_dijkstra(), bellman_ford_path() |
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155 single_source_dijkstra() |
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156 """ |
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157 (length, path) = single_source_dijkstra(G, source, target=target, weight=weight) |
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158 return path |
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159 |
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160 |
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161 def dijkstra_path_length(G, source, target, weight="weight"): |
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162 """Returns the shortest weighted path length in G from source to target. |
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163 |
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164 Uses Dijkstra's Method to compute the shortest weighted path length |
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165 between two nodes in a graph. |
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166 |
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167 Parameters |
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168 ---------- |
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169 G : NetworkX graph |
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170 |
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171 source : node label |
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172 starting node for path |
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173 |
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174 target : node label |
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175 ending node for path |
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176 |
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177 weight : string or function |
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178 If this is a string, then edge weights will be accessed via the |
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179 edge attribute with this key (that is, the weight of the edge |
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180 joining `u` to `v` will be ``G.edges[u, v][weight]``). If no |
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181 such edge attribute exists, the weight of the edge is assumed to |
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182 be one. |
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183 |
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184 If this is a function, the weight of an edge is the value |
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185 returned by the function. The function must accept exactly three |
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186 positional arguments: the two endpoints of an edge and the |
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187 dictionary of edge attributes for that edge. The function must |
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188 return a number. |
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189 |
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190 Returns |
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191 ------- |
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192 length : number |
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193 Shortest path length. |
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194 |
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195 Raises |
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196 ------ |
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197 NodeNotFound |
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198 If `source` is not in `G`. |
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199 |
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200 NetworkXNoPath |
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201 If no path exists between source and target. |
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202 |
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203 Examples |
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204 -------- |
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205 >>> G = nx.path_graph(5) |
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206 >>> print(nx.dijkstra_path_length(G, 0, 4)) |
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207 4 |
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208 |
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209 Notes |
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210 ----- |
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211 Edge weight attributes must be numerical. |
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212 Distances are calculated as sums of weighted edges traversed. |
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213 |
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214 The weight function can be used to hide edges by returning None. |
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215 So ``weight = lambda u, v, d: 1 if d['color']=="red" else None`` |
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216 will find the shortest red path. |
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217 |
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218 The function :func:`single_source_dijkstra` computes both |
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219 path and length-of-path if you need both, use that. |
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220 |
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221 See Also |
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222 -------- |
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223 bidirectional_dijkstra(), bellman_ford_path_length() |
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224 single_source_dijkstra() |
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225 |
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226 """ |
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227 if source == target: |
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228 return 0 |
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229 weight = _weight_function(G, weight) |
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230 length = _dijkstra(G, source, weight, target=target) |
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231 try: |
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232 return length[target] |
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233 except KeyError as e: |
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234 raise nx.NetworkXNoPath(f"Node {target} not reachable from {source}") from e |
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235 |
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236 |
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237 def single_source_dijkstra_path(G, source, cutoff=None, weight="weight"): |
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238 """Find shortest weighted paths in G from a source node. |
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239 |
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240 Compute shortest path between source and all other reachable |
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241 nodes for a weighted graph. |
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242 |
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243 Parameters |
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244 ---------- |
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245 G : NetworkX graph |
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246 |
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247 source : node |
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248 Starting node for path. |
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249 |
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250 cutoff : integer or float, optional |
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251 Depth to stop the search. Only return paths with length <= cutoff. |
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252 |
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253 weight : string or function |
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254 If this is a string, then edge weights will be accessed via the |
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255 edge attribute with this key (that is, the weight of the edge |
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256 joining `u` to `v` will be ``G.edges[u, v][weight]``). If no |
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257 such edge attribute exists, the weight of the edge is assumed to |
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258 be one. |
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259 |
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260 If this is a function, the weight of an edge is the value |
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261 returned by the function. The function must accept exactly three |
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262 positional arguments: the two endpoints of an edge and the |
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263 dictionary of edge attributes for that edge. The function must |
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264 return a number. |
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265 |
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266 Returns |
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267 ------- |
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268 paths : dictionary |
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269 Dictionary of shortest path lengths keyed by target. |
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270 |
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271 Raises |
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272 ------ |
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273 NodeNotFound |
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274 If `source` is not in `G`. |
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275 |
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276 Examples |
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277 -------- |
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278 >>> G = nx.path_graph(5) |
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279 >>> path = nx.single_source_dijkstra_path(G, 0) |
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280 >>> path[4] |
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281 [0, 1, 2, 3, 4] |
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282 |
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283 Notes |
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284 ----- |
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285 Edge weight attributes must be numerical. |
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286 Distances are calculated as sums of weighted edges traversed. |
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287 |
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288 The weight function can be used to hide edges by returning None. |
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289 So ``weight = lambda u, v, d: 1 if d['color']=="red" else None`` |
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290 will find the shortest red path. |
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291 |
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292 See Also |
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293 -------- |
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294 single_source_dijkstra(), single_source_bellman_ford() |
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295 |
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296 """ |
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297 return multi_source_dijkstra_path(G, {source}, cutoff=cutoff, weight=weight) |
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298 |
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299 |
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300 def single_source_dijkstra_path_length(G, source, cutoff=None, weight="weight"): |
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301 """Find shortest weighted path lengths in G from a source node. |
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302 |
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303 Compute the shortest path length between source and all other |
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304 reachable nodes for a weighted graph. |
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305 |
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306 Parameters |
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307 ---------- |
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308 G : NetworkX graph |
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309 |
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310 source : node label |
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311 Starting node for path |
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312 |
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313 cutoff : integer or float, optional |
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314 Depth to stop the search. Only return paths with length <= cutoff. |
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315 |
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316 weight : string or function |
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317 If this is a string, then edge weights will be accessed via the |
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318 edge attribute with this key (that is, the weight of the edge |
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319 joining `u` to `v` will be ``G.edges[u, v][weight]``). If no |
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320 such edge attribute exists, the weight of the edge is assumed to |
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321 be one. |
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322 |
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323 If this is a function, the weight of an edge is the value |
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324 returned by the function. The function must accept exactly three |
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325 positional arguments: the two endpoints of an edge and the |
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326 dictionary of edge attributes for that edge. The function must |
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327 return a number. |
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328 |
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329 Returns |
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330 ------- |
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331 length : dict |
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332 Dict keyed by node to shortest path length from source. |
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333 |
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334 Raises |
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335 ------ |
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336 NodeNotFound |
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337 If `source` is not in `G`. |
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338 |
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339 Examples |
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340 -------- |
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341 >>> G = nx.path_graph(5) |
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342 >>> length = nx.single_source_dijkstra_path_length(G, 0) |
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343 >>> length[4] |
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344 4 |
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345 >>> for node in [0, 1, 2, 3, 4]: |
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346 ... print(f"{node}: {length[node]}") |
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347 0: 0 |
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348 1: 1 |
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349 2: 2 |
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350 3: 3 |
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351 4: 4 |
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352 |
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353 Notes |
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354 ----- |
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355 Edge weight attributes must be numerical. |
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356 Distances are calculated as sums of weighted edges traversed. |
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357 |
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358 The weight function can be used to hide edges by returning None. |
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359 So ``weight = lambda u, v, d: 1 if d['color']=="red" else None`` |
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360 will find the shortest red path. |
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361 |
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362 See Also |
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363 -------- |
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364 single_source_dijkstra(), single_source_bellman_ford_path_length() |
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365 |
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366 """ |
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367 return multi_source_dijkstra_path_length(G, {source}, cutoff=cutoff, weight=weight) |
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368 |
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369 |
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370 def single_source_dijkstra(G, source, target=None, cutoff=None, weight="weight"): |
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371 """Find shortest weighted paths and lengths from a source node. |
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372 |
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373 Compute the shortest path length between source and all other |
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374 reachable nodes for a weighted graph. |
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375 |
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376 Uses Dijkstra's algorithm to compute shortest paths and lengths |
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377 between a source and all other reachable nodes in a weighted graph. |
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378 |
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379 Parameters |
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380 ---------- |
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381 G : NetworkX graph |
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382 |
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383 source : node label |
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384 Starting node for path |
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385 |
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386 target : node label, optional |
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387 Ending node for path |
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388 |
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389 cutoff : integer or float, optional |
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390 Depth to stop the search. Only return paths with length <= cutoff. |
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391 |
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392 weight : string or function |
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393 If this is a string, then edge weights will be accessed via the |
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394 edge attribute with this key (that is, the weight of the edge |
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395 joining `u` to `v` will be ``G.edges[u, v][weight]``). If no |
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396 such edge attribute exists, the weight of the edge is assumed to |
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397 be one. |
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398 |
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399 If this is a function, the weight of an edge is the value |
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400 returned by the function. The function must accept exactly three |
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401 positional arguments: the two endpoints of an edge and the |
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402 dictionary of edge attributes for that edge. The function must |
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403 return a number. |
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404 |
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405 Returns |
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406 ------- |
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407 distance, path : pair of dictionaries, or numeric and list. |
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408 If target is None, paths and lengths to all nodes are computed. |
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409 The return value is a tuple of two dictionaries keyed by target nodes. |
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410 The first dictionary stores distance to each target node. |
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411 The second stores the path to each target node. |
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412 If target is not None, returns a tuple (distance, path), where |
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413 distance is the distance from source to target and path is a list |
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414 representing the path from source to target. |
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415 |
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416 Raises |
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417 ------ |
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418 NodeNotFound |
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419 If `source` is not in `G`. |
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420 |
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421 Examples |
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422 -------- |
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423 >>> G = nx.path_graph(5) |
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424 >>> length, path = nx.single_source_dijkstra(G, 0) |
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425 >>> print(length[4]) |
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426 4 |
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427 >>> for node in [0, 1, 2, 3, 4]: |
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428 ... print(f"{node}: {length[node]}") |
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429 0: 0 |
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430 1: 1 |
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431 2: 2 |
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432 3: 3 |
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433 4: 4 |
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434 >>> path[4] |
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435 [0, 1, 2, 3, 4] |
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436 >>> length, path = nx.single_source_dijkstra(G, 0, 1) |
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437 >>> length |
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438 1 |
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439 >>> path |
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440 [0, 1] |
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441 |
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442 Notes |
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443 ----- |
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444 Edge weight attributes must be numerical. |
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445 Distances are calculated as sums of weighted edges traversed. |
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446 |
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447 The weight function can be used to hide edges by returning None. |
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448 So ``weight = lambda u, v, d: 1 if d['color']=="red" else None`` |
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449 will find the shortest red path. |
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450 |
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451 Based on the Python cookbook recipe (119466) at |
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452 http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/119466 |
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453 |
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454 This algorithm is not guaranteed to work if edge weights |
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455 are negative or are floating point numbers |
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456 (overflows and roundoff errors can cause problems). |
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457 |
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458 See Also |
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459 -------- |
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460 single_source_dijkstra_path() |
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461 single_source_dijkstra_path_length() |
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462 single_source_bellman_ford() |
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463 """ |
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464 return multi_source_dijkstra( |
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465 G, {source}, cutoff=cutoff, target=target, weight=weight |
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466 ) |
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467 |
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468 |
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469 def multi_source_dijkstra_path(G, sources, cutoff=None, weight="weight"): |
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470 """Find shortest weighted paths in G from a given set of source |
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471 nodes. |
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472 |
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473 Compute shortest path between any of the source nodes and all other |
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474 reachable nodes for a weighted graph. |
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475 |
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476 Parameters |
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477 ---------- |
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478 G : NetworkX graph |
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479 |
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480 sources : non-empty set of nodes |
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481 Starting nodes for paths. If this is just a set containing a |
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482 single node, then all paths computed by this function will start |
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483 from that node. If there are two or more nodes in the set, the |
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484 computed paths may begin from any one of the start nodes. |
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485 |
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486 cutoff : integer or float, optional |
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487 Depth to stop the search. Only return paths with length <= cutoff. |
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488 |
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489 weight : string or function |
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490 If this is a string, then edge weights will be accessed via the |
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491 edge attribute with this key (that is, the weight of the edge |
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492 joining `u` to `v` will be ``G.edges[u, v][weight]``). If no |
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493 such edge attribute exists, the weight of the edge is assumed to |
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494 be one. |
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495 |
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496 If this is a function, the weight of an edge is the value |
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497 returned by the function. The function must accept exactly three |
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498 positional arguments: the two endpoints of an edge and the |
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499 dictionary of edge attributes for that edge. The function must |
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500 return a number. |
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501 |
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502 Returns |
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503 ------- |
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504 paths : dictionary |
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505 Dictionary of shortest paths keyed by target. |
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506 |
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507 Examples |
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508 -------- |
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509 >>> G = nx.path_graph(5) |
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510 >>> path = nx.multi_source_dijkstra_path(G, {0, 4}) |
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511 >>> path[1] |
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512 [0, 1] |
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513 >>> path[3] |
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514 [4, 3] |
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515 |
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516 Notes |
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517 ----- |
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518 Edge weight attributes must be numerical. |
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519 Distances are calculated as sums of weighted edges traversed. |
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520 |
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521 The weight function can be used to hide edges by returning None. |
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522 So ``weight = lambda u, v, d: 1 if d['color']=="red" else None`` |
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523 will find the shortest red path. |
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524 |
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525 Raises |
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526 ------ |
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527 ValueError |
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528 If `sources` is empty. |
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529 NodeNotFound |
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530 If any of `sources` is not in `G`. |
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531 |
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532 See Also |
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533 -------- |
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534 multi_source_dijkstra(), multi_source_bellman_ford() |
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535 |
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536 """ |
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537 length, path = multi_source_dijkstra(G, sources, cutoff=cutoff, weight=weight) |
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538 return path |
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539 |
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540 |
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541 def multi_source_dijkstra_path_length(G, sources, cutoff=None, weight="weight"): |
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542 """Find shortest weighted path lengths in G from a given set of |
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543 source nodes. |
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544 |
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545 Compute the shortest path length between any of the source nodes and |
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546 all other reachable nodes for a weighted graph. |
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547 |
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548 Parameters |
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549 ---------- |
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550 G : NetworkX graph |
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551 |
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552 sources : non-empty set of nodes |
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553 Starting nodes for paths. If this is just a set containing a |
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554 single node, then all paths computed by this function will start |
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555 from that node. If there are two or more nodes in the set, the |
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556 computed paths may begin from any one of the start nodes. |
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557 |
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558 cutoff : integer or float, optional |
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559 Depth to stop the search. Only return paths with length <= cutoff. |
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560 |
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561 weight : string or function |
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562 If this is a string, then edge weights will be accessed via the |
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563 edge attribute with this key (that is, the weight of the edge |
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564 joining `u` to `v` will be ``G.edges[u, v][weight]``). If no |
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565 such edge attribute exists, the weight of the edge is assumed to |
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566 be one. |
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567 |
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568 If this is a function, the weight of an edge is the value |
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569 returned by the function. The function must accept exactly three |
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570 positional arguments: the two endpoints of an edge and the |
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571 dictionary of edge attributes for that edge. The function must |
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572 return a number. |
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573 |
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574 Returns |
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575 ------- |
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576 length : dict |
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577 Dict keyed by node to shortest path length to nearest source. |
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578 |
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579 Examples |
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580 -------- |
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581 >>> G = nx.path_graph(5) |
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582 >>> length = nx.multi_source_dijkstra_path_length(G, {0, 4}) |
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583 >>> for node in [0, 1, 2, 3, 4]: |
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584 ... print(f"{node}: {length[node]}") |
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585 0: 0 |
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586 1: 1 |
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587 2: 2 |
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588 3: 1 |
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589 4: 0 |
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590 |
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591 Notes |
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592 ----- |
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593 Edge weight attributes must be numerical. |
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594 Distances are calculated as sums of weighted edges traversed. |
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595 |
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596 The weight function can be used to hide edges by returning None. |
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597 So ``weight = lambda u, v, d: 1 if d['color']=="red" else None`` |
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598 will find the shortest red path. |
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599 |
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600 Raises |
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601 ------ |
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602 ValueError |
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603 If `sources` is empty. |
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604 NodeNotFound |
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605 If any of `sources` is not in `G`. |
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606 |
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607 See Also |
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608 -------- |
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609 multi_source_dijkstra() |
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610 |
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611 """ |
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612 if not sources: |
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613 raise ValueError("sources must not be empty") |
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614 weight = _weight_function(G, weight) |
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615 return _dijkstra_multisource(G, sources, weight, cutoff=cutoff) |
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616 |
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617 |
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618 def multi_source_dijkstra(G, sources, target=None, cutoff=None, weight="weight"): |
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619 """Find shortest weighted paths and lengths from a given set of |
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620 source nodes. |
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621 |
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622 Uses Dijkstra's algorithm to compute the shortest paths and lengths |
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623 between one of the source nodes and the given `target`, or all other |
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624 reachable nodes if not specified, for a weighted graph. |
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625 |
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626 Parameters |
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627 ---------- |
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628 G : NetworkX graph |
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629 |
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630 sources : non-empty set of nodes |
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631 Starting nodes for paths. If this is just a set containing a |
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632 single node, then all paths computed by this function will start |
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633 from that node. If there are two or more nodes in the set, the |
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634 computed paths may begin from any one of the start nodes. |
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635 |
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636 target : node label, optional |
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637 Ending node for path |
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638 |
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639 cutoff : integer or float, optional |
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640 Depth to stop the search. Only return paths with length <= cutoff. |
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641 |
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642 weight : string or function |
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643 If this is a string, then edge weights will be accessed via the |
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644 edge attribute with this key (that is, the weight of the edge |
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645 joining `u` to `v` will be ``G.edges[u, v][weight]``). If no |
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646 such edge attribute exists, the weight of the edge is assumed to |
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647 be one. |
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648 |
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649 If this is a function, the weight of an edge is the value |
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650 returned by the function. The function must accept exactly three |
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651 positional arguments: the two endpoints of an edge and the |
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652 dictionary of edge attributes for that edge. The function must |
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653 return a number. |
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654 |
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655 Returns |
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656 ------- |
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657 distance, path : pair of dictionaries, or numeric and list |
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658 If target is None, returns a tuple of two dictionaries keyed by node. |
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659 The first dictionary stores distance from one of the source nodes. |
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660 The second stores the path from one of the sources to that node. |
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661 If target is not None, returns a tuple of (distance, path) where |
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662 distance is the distance from source to target and path is a list |
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663 representing the path from source to target. |
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664 |
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665 Examples |
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666 -------- |
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667 >>> G = nx.path_graph(5) |
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668 >>> length, path = nx.multi_source_dijkstra(G, {0, 4}) |
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669 >>> for node in [0, 1, 2, 3, 4]: |
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670 ... print(f"{node}: {length[node]}") |
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671 0: 0 |
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672 1: 1 |
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673 2: 2 |
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674 3: 1 |
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675 4: 0 |
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676 >>> path[1] |
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677 [0, 1] |
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678 >>> path[3] |
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679 [4, 3] |
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680 |
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681 >>> length, path = nx.multi_source_dijkstra(G, {0, 4}, 1) |
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682 >>> length |
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683 1 |
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684 >>> path |
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685 [0, 1] |
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686 |
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687 Notes |
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688 ----- |
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689 Edge weight attributes must be numerical. |
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690 Distances are calculated as sums of weighted edges traversed. |
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691 |
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692 The weight function can be used to hide edges by returning None. |
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693 So ``weight = lambda u, v, d: 1 if d['color']=="red" else None`` |
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694 will find the shortest red path. |
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695 |
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696 Based on the Python cookbook recipe (119466) at |
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697 http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/119466 |
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698 |
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699 This algorithm is not guaranteed to work if edge weights |
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700 are negative or are floating point numbers |
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701 (overflows and roundoff errors can cause problems). |
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702 |
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703 Raises |
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704 ------ |
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705 ValueError |
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706 If `sources` is empty. |
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707 NodeNotFound |
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708 If any of `sources` is not in `G`. |
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709 |
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710 See Also |
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711 -------- |
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712 multi_source_dijkstra_path() |
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713 multi_source_dijkstra_path_length() |
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714 |
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715 """ |
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716 if not sources: |
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717 raise ValueError("sources must not be empty") |
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718 if target in sources: |
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719 return (0, [target]) |
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720 weight = _weight_function(G, weight) |
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721 paths = {source: [source] for source in sources} # dictionary of paths |
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722 dist = _dijkstra_multisource( |
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723 G, sources, weight, paths=paths, cutoff=cutoff, target=target |
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724 ) |
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725 if target is None: |
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726 return (dist, paths) |
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727 try: |
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728 return (dist[target], paths[target]) |
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729 except KeyError as e: |
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730 raise nx.NetworkXNoPath(f"No path to {target}.") from e |
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731 |
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732 |
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733 def _dijkstra(G, source, weight, pred=None, paths=None, cutoff=None, target=None): |
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734 """Uses Dijkstra's algorithm to find shortest weighted paths from a |
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735 single source. |
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736 |
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737 This is a convenience function for :func:`_dijkstra_multisource` |
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738 with all the arguments the same, except the keyword argument |
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739 `sources` set to ``[source]``. |
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740 |
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741 """ |
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742 return _dijkstra_multisource( |
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743 G, [source], weight, pred=pred, paths=paths, cutoff=cutoff, target=target |
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744 ) |
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745 |
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746 |
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747 def _dijkstra_multisource( |
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748 G, sources, weight, pred=None, paths=None, cutoff=None, target=None |
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749 ): |
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750 """Uses Dijkstra's algorithm to find shortest weighted paths |
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751 |
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752 Parameters |
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753 ---------- |
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754 G : NetworkX graph |
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755 |
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756 sources : non-empty iterable of nodes |
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757 Starting nodes for paths. If this is just an iterable containing |
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758 a single node, then all paths computed by this function will |
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759 start from that node. If there are two or more nodes in this |
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760 iterable, the computed paths may begin from any one of the start |
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761 nodes. |
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762 |
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763 weight: function |
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764 Function with (u, v, data) input that returns that edges weight |
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765 |
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766 pred: dict of lists, optional(default=None) |
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767 dict to store a list of predecessors keyed by that node |
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768 If None, predecessors are not stored. |
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769 |
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770 paths: dict, optional (default=None) |
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771 dict to store the path list from source to each node, keyed by node. |
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772 If None, paths are not stored. |
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773 |
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774 target : node label, optional |
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775 Ending node for path. Search is halted when target is found. |
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776 |
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777 cutoff : integer or float, optional |
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778 Depth to stop the search. Only return paths with length <= cutoff. |
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779 |
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780 Returns |
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781 ------- |
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782 distance : dictionary |
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783 A mapping from node to shortest distance to that node from one |
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784 of the source nodes. |
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785 |
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786 Raises |
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787 ------ |
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788 NodeNotFound |
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789 If any of `sources` is not in `G`. |
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790 |
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791 Notes |
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792 ----- |
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793 The optional predecessor and path dictionaries can be accessed by |
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794 the caller through the original pred and paths objects passed |
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795 as arguments. No need to explicitly return pred or paths. |
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796 |
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797 """ |
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798 G_succ = G._succ if G.is_directed() else G._adj |
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799 |
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800 push = heappush |
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801 pop = heappop |
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802 dist = {} # dictionary of final distances |
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803 seen = {} |
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804 # fringe is heapq with 3-tuples (distance,c,node) |
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805 # use the count c to avoid comparing nodes (may not be able to) |
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806 c = count() |
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807 fringe = [] |
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808 for source in sources: |
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809 if source not in G: |
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810 raise nx.NodeNotFound(f"Source {source} not in G") |
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811 seen[source] = 0 |
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812 push(fringe, (0, next(c), source)) |
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813 while fringe: |
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814 (d, _, v) = pop(fringe) |
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815 if v in dist: |
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816 continue # already searched this node. |
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817 dist[v] = d |
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818 if v == target: |
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819 break |
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820 for u, e in G_succ[v].items(): |
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821 cost = weight(v, u, e) |
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822 if cost is None: |
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823 continue |
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824 vu_dist = dist[v] + cost |
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825 if cutoff is not None: |
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826 if vu_dist > cutoff: |
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827 continue |
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828 if u in dist: |
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829 u_dist = dist[u] |
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830 if vu_dist < u_dist: |
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831 raise ValueError("Contradictory paths found:", "negative weights?") |
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832 elif pred is not None and vu_dist == u_dist: |
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833 pred[u].append(v) |
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834 elif u not in seen or vu_dist < seen[u]: |
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835 seen[u] = vu_dist |
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836 push(fringe, (vu_dist, next(c), u)) |
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837 if paths is not None: |
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838 paths[u] = paths[v] + [u] |
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839 if pred is not None: |
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840 pred[u] = [v] |
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841 elif vu_dist == seen[u]: |
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842 if pred is not None: |
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843 pred[u].append(v) |
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844 |
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845 # The optional predecessor and path dictionaries can be accessed |
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846 # by the caller via the pred and paths objects passed as arguments. |
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847 return dist |
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848 |
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849 |
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850 def dijkstra_predecessor_and_distance(G, source, cutoff=None, weight="weight"): |
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851 """Compute weighted shortest path length and predecessors. |
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852 |
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853 Uses Dijkstra's Method to obtain the shortest weighted paths |
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854 and return dictionaries of predecessors for each node and |
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855 distance for each node from the `source`. |
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856 |
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857 Parameters |
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858 ---------- |
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859 G : NetworkX graph |
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860 |
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861 source : node label |
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862 Starting node for path |
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863 |
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864 cutoff : integer or float, optional |
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865 Depth to stop the search. Only return paths with length <= cutoff. |
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866 |
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867 weight : string or function |
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868 If this is a string, then edge weights will be accessed via the |
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869 edge attribute with this key (that is, the weight of the edge |
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870 joining `u` to `v` will be ``G.edges[u, v][weight]``). If no |
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871 such edge attribute exists, the weight of the edge is assumed to |
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872 be one. |
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873 |
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874 If this is a function, the weight of an edge is the value |
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875 returned by the function. The function must accept exactly three |
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876 positional arguments: the two endpoints of an edge and the |
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877 dictionary of edge attributes for that edge. The function must |
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878 return a number. |
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879 |
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880 Returns |
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881 ------- |
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882 pred, distance : dictionaries |
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883 Returns two dictionaries representing a list of predecessors |
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884 of a node and the distance to each node. |
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885 Warning: If target is specified, the dicts are incomplete as they |
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886 only contain information for the nodes along a path to target. |
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887 |
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888 Raises |
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889 ------ |
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890 NodeNotFound |
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891 If `source` is not in `G`. |
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892 |
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893 Notes |
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894 ----- |
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895 Edge weight attributes must be numerical. |
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896 Distances are calculated as sums of weighted edges traversed. |
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897 |
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898 The list of predecessors contains more than one element only when |
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899 there are more than one shortest paths to the key node. |
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900 |
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901 Examples |
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902 -------- |
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903 >>> G = nx.path_graph(5, create_using=nx.DiGraph()) |
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904 >>> pred, dist = nx.dijkstra_predecessor_and_distance(G, 0) |
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905 >>> sorted(pred.items()) |
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906 [(0, []), (1, [0]), (2, [1]), (3, [2]), (4, [3])] |
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907 >>> sorted(dist.items()) |
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908 [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)] |
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909 |
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910 >>> pred, dist = nx.dijkstra_predecessor_and_distance(G, 0, 1) |
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911 >>> sorted(pred.items()) |
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912 [(0, []), (1, [0])] |
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913 >>> sorted(dist.items()) |
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914 [(0, 0), (1, 1)] |
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915 """ |
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916 |
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917 weight = _weight_function(G, weight) |
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918 pred = {source: []} # dictionary of predecessors |
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919 return (pred, _dijkstra(G, source, weight, pred=pred, cutoff=cutoff)) |
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920 |
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921 |
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922 def all_pairs_dijkstra(G, cutoff=None, weight="weight"): |
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923 """Find shortest weighted paths and lengths between all nodes. |
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924 |
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925 Parameters |
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926 ---------- |
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927 G : NetworkX graph |
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928 |
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929 cutoff : integer or float, optional |
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930 Depth to stop the search. Only return paths with length <= cutoff. |
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931 |
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932 weight : string or function |
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933 If this is a string, then edge weights will be accessed via the |
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934 edge attribute with this key (that is, the weight of the edge |
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935 joining `u` to `v` will be ``G.edge[u][v][weight]``). If no |
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936 such edge attribute exists, the weight of the edge is assumed to |
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937 be one. |
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938 |
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939 If this is a function, the weight of an edge is the value |
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940 returned by the function. The function must accept exactly three |
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941 positional arguments: the two endpoints of an edge and the |
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942 dictionary of edge attributes for that edge. The function must |
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943 return a number. |
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944 |
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945 Yields |
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946 ------ |
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947 (node, (distance, path)) : (node obj, (dict, dict)) |
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948 Each source node has two associated dicts. The first holds distance |
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949 keyed by target and the second holds paths keyed by target. |
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950 (See single_source_dijkstra for the source/target node terminology.) |
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951 If desired you can apply `dict()` to this function to create a dict |
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952 keyed by source node to the two dicts. |
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953 |
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954 Examples |
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955 -------- |
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956 >>> G = nx.path_graph(5) |
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957 >>> len_path = dict(nx.all_pairs_dijkstra(G)) |
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958 >>> print(len_path[3][0][1]) |
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959 2 |
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960 >>> for node in [0, 1, 2, 3, 4]: |
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961 ... print(f"3 - {node}: {len_path[3][0][node]}") |
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962 3 - 0: 3 |
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963 3 - 1: 2 |
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964 3 - 2: 1 |
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965 3 - 3: 0 |
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966 3 - 4: 1 |
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967 >>> len_path[3][1][1] |
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968 [3, 2, 1] |
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969 >>> for n, (dist, path) in nx.all_pairs_dijkstra(G): |
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970 ... print(path[1]) |
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971 [0, 1] |
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972 [1] |
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973 [2, 1] |
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974 [3, 2, 1] |
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975 [4, 3, 2, 1] |
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976 |
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977 Notes |
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978 ----- |
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979 Edge weight attributes must be numerical. |
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980 Distances are calculated as sums of weighted edges traversed. |
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981 |
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982 The yielded dicts only have keys for reachable nodes. |
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983 """ |
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984 for n in G: |
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985 dist, path = single_source_dijkstra(G, n, cutoff=cutoff, weight=weight) |
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986 yield (n, (dist, path)) |
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987 |
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988 |
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989 def all_pairs_dijkstra_path_length(G, cutoff=None, weight="weight"): |
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990 """Compute shortest path lengths between all nodes in a weighted graph. |
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991 |
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992 Parameters |
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993 ---------- |
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994 G : NetworkX graph |
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995 |
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996 cutoff : integer or float, optional |
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997 Depth to stop the search. Only return paths with length <= cutoff. |
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998 |
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999 weight : string or function |
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1000 If this is a string, then edge weights will be accessed via the |
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1001 edge attribute with this key (that is, the weight of the edge |
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1002 joining `u` to `v` will be ``G.edges[u, v][weight]``). If no |
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1003 such edge attribute exists, the weight of the edge is assumed to |
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1004 be one. |
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1005 |
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1006 If this is a function, the weight of an edge is the value |
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1007 returned by the function. The function must accept exactly three |
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1008 positional arguments: the two endpoints of an edge and the |
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1009 dictionary of edge attributes for that edge. The function must |
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1010 return a number. |
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1011 |
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1012 Returns |
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1013 ------- |
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1014 distance : iterator |
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1015 (source, dictionary) iterator with dictionary keyed by target and |
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1016 shortest path length as the key value. |
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1017 |
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1018 Examples |
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1019 -------- |
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1020 >>> G = nx.path_graph(5) |
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1021 >>> length = dict(nx.all_pairs_dijkstra_path_length(G)) |
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1022 >>> for node in [0, 1, 2, 3, 4]: |
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1023 ... print(f"1 - {node}: {length[1][node]}") |
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1024 1 - 0: 1 |
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1025 1 - 1: 0 |
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1026 1 - 2: 1 |
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1027 1 - 3: 2 |
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1028 1 - 4: 3 |
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1029 >>> length[3][2] |
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1030 1 |
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1031 >>> length[2][2] |
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1032 0 |
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1033 |
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1034 Notes |
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1035 ----- |
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1036 Edge weight attributes must be numerical. |
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1037 Distances are calculated as sums of weighted edges traversed. |
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1038 |
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1039 The dictionary returned only has keys for reachable node pairs. |
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1040 """ |
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1041 length = single_source_dijkstra_path_length |
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1042 for n in G: |
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1043 yield (n, length(G, n, cutoff=cutoff, weight=weight)) |
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1044 |
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1045 |
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1046 def all_pairs_dijkstra_path(G, cutoff=None, weight="weight"): |
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1047 """Compute shortest paths between all nodes in a weighted graph. |
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1048 |
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1049 Parameters |
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1050 ---------- |
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1051 G : NetworkX graph |
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1052 |
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1053 cutoff : integer or float, optional |
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1054 Depth to stop the search. Only return paths with length <= cutoff. |
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1055 |
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1056 weight : string or function |
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1057 If this is a string, then edge weights will be accessed via the |
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1058 edge attribute with this key (that is, the weight of the edge |
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1059 joining `u` to `v` will be ``G.edges[u, v][weight]``). If no |
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1060 such edge attribute exists, the weight of the edge is assumed to |
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1061 be one. |
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1062 |
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1063 If this is a function, the weight of an edge is the value |
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1064 returned by the function. The function must accept exactly three |
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1065 positional arguments: the two endpoints of an edge and the |
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1066 dictionary of edge attributes for that edge. The function must |
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1067 return a number. |
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1068 |
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1069 Returns |
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1070 ------- |
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1071 distance : dictionary |
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1072 Dictionary, keyed by source and target, of shortest paths. |
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1073 |
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1074 Examples |
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1075 -------- |
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1076 >>> G = nx.path_graph(5) |
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1077 >>> path = dict(nx.all_pairs_dijkstra_path(G)) |
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1078 >>> print(path[0][4]) |
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1079 [0, 1, 2, 3, 4] |
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1080 |
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1081 Notes |
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1082 ----- |
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1083 Edge weight attributes must be numerical. |
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1084 Distances are calculated as sums of weighted edges traversed. |
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1085 |
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1086 See Also |
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1087 -------- |
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1088 floyd_warshall(), all_pairs_bellman_ford_path() |
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1089 |
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1090 """ |
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1091 path = single_source_dijkstra_path |
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1092 # TODO This can be trivially parallelized. |
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1093 for n in G: |
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1094 yield (n, path(G, n, cutoff=cutoff, weight=weight)) |
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1095 |
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1096 |
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1097 def bellman_ford_predecessor_and_distance( |
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1098 G, source, target=None, weight="weight", heuristic=False |
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1099 ): |
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1100 """Compute shortest path lengths and predecessors on shortest paths |
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1101 in weighted graphs. |
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1102 |
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1103 The algorithm has a running time of $O(mn)$ where $n$ is the number of |
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1104 nodes and $m$ is the number of edges. It is slower than Dijkstra but |
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1105 can handle negative edge weights. |
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1106 |
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1107 Parameters |
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1108 ---------- |
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1109 G : NetworkX graph |
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1110 The algorithm works for all types of graphs, including directed |
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1111 graphs and multigraphs. |
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1112 |
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1113 source: node label |
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1114 Starting node for path |
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1115 |
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1116 weight : string or function |
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1117 If this is a string, then edge weights will be accessed via the |
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1118 edge attribute with this key (that is, the weight of the edge |
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1119 joining `u` to `v` will be ``G.edges[u, v][weight]``). If no |
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1120 such edge attribute exists, the weight of the edge is assumed to |
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1121 be one. |
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1122 |
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1123 If this is a function, the weight of an edge is the value |
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1124 returned by the function. The function must accept exactly three |
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1125 positional arguments: the two endpoints of an edge and the |
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1126 dictionary of edge attributes for that edge. The function must |
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1127 return a number. |
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1128 |
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1129 heuristic : bool |
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1130 Determines whether to use a heuristic to early detect negative |
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1131 cycles at a hopefully negligible cost. |
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1132 |
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1133 Returns |
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1134 ------- |
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1135 pred, dist : dictionaries |
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1136 Returns two dictionaries keyed by node to predecessor in the |
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1137 path and to the distance from the source respectively. |
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1138 |
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1139 Raises |
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1140 ------ |
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1141 NodeNotFound |
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1142 If `source` is not in `G`. |
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1143 |
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1144 NetworkXUnbounded |
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1145 If the (di)graph contains a negative cost (di)cycle, the |
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1146 algorithm raises an exception to indicate the presence of the |
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1147 negative cost (di)cycle. Note: any negative weight edge in an |
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1148 undirected graph is a negative cost cycle. |
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1149 |
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1150 Examples |
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1151 -------- |
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1152 >>> G = nx.path_graph(5, create_using=nx.DiGraph()) |
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1153 >>> pred, dist = nx.bellman_ford_predecessor_and_distance(G, 0) |
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1154 >>> sorted(pred.items()) |
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1155 [(0, []), (1, [0]), (2, [1]), (3, [2]), (4, [3])] |
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1156 >>> sorted(dist.items()) |
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1157 [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)] |
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1158 |
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1159 >>> pred, dist = nx.bellman_ford_predecessor_and_distance(G, 0, 1) |
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1160 >>> sorted(pred.items()) |
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1161 [(0, []), (1, [0]), (2, [1]), (3, [2]), (4, [3])] |
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1162 >>> sorted(dist.items()) |
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1163 [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)] |
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1164 |
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1165 >>> G = nx.cycle_graph(5, create_using=nx.DiGraph()) |
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1166 >>> G[1][2]["weight"] = -7 |
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1167 >>> nx.bellman_ford_predecessor_and_distance(G, 0) |
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1168 Traceback (most recent call last): |
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1169 ... |
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1170 networkx.exception.NetworkXUnbounded: Negative cost cycle detected. |
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1171 |
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1172 Notes |
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1173 ----- |
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1174 Edge weight attributes must be numerical. |
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1175 Distances are calculated as sums of weighted edges traversed. |
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1176 |
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1177 The dictionaries returned only have keys for nodes reachable from |
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1178 the source. |
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1179 |
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1180 In the case where the (di)graph is not connected, if a component |
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1181 not containing the source contains a negative cost (di)cycle, it |
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1182 will not be detected. |
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1183 |
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1184 In NetworkX v2.1 and prior, the source node had predecessor `[None]`. |
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1185 In NetworkX v2.2 this changed to the source node having predecessor `[]` |
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1186 """ |
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1187 if source not in G: |
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1188 raise nx.NodeNotFound(f"Node {source} is not found in the graph") |
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1189 weight = _weight_function(G, weight) |
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1190 if any(weight(u, v, d) < 0 for u, v, d in nx.selfloop_edges(G, data=True)): |
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1191 raise nx.NetworkXUnbounded("Negative cost cycle detected.") |
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1192 |
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1193 dist = {source: 0} |
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1194 pred = {source: []} |
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1195 |
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1196 if len(G) == 1: |
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1197 return pred, dist |
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1198 |
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1199 weight = _weight_function(G, weight) |
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1200 |
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1201 dist = _bellman_ford( |
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1202 G, [source], weight, pred=pred, dist=dist, target=target, heuristic=heuristic |
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1203 ) |
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1204 return (pred, dist) |
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1205 |
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1206 |
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1207 def _bellman_ford( |
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1208 G, source, weight, pred=None, paths=None, dist=None, target=None, heuristic=True |
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1209 ): |
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1210 """Relaxation loop for Bellman–Ford algorithm. |
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1211 |
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1212 This is an implementation of the SPFA variant. |
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1213 See https://en.wikipedia.org/wiki/Shortest_Path_Faster_Algorithm |
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1214 |
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1215 Parameters |
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1216 ---------- |
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1217 G : NetworkX graph |
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1218 |
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1219 source: list |
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1220 List of source nodes. The shortest path from any of the source |
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1221 nodes will be found if multiple sources are provided. |
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1222 |
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1223 weight : function |
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1224 The weight of an edge is the value returned by the function. The |
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1225 function must accept exactly three positional arguments: the two |
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1226 endpoints of an edge and the dictionary of edge attributes for |
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1227 that edge. The function must return a number. |
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1228 |
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1229 pred: dict of lists, optional (default=None) |
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1230 dict to store a list of predecessors keyed by that node |
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1231 If None, predecessors are not stored |
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1232 |
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1233 paths: dict, optional (default=None) |
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1234 dict to store the path list from source to each node, keyed by node |
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1235 If None, paths are not stored |
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1236 |
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1237 dist: dict, optional (default=None) |
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1238 dict to store distance from source to the keyed node |
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1239 If None, returned dist dict contents default to 0 for every node in the |
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1240 source list |
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1241 |
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1242 target: node label, optional |
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1243 Ending node for path. Path lengths to other destinations may (and |
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1244 probably will) be incorrect. |
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1245 |
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1246 heuristic : bool |
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1247 Determines whether to use a heuristic to early detect negative |
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1248 cycles at a hopefully negligible cost. |
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1249 |
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1250 Returns |
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1251 ------- |
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1252 Returns a dict keyed by node to the distance from the source. |
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1253 Dicts for paths and pred are in the mutated input dicts by those names. |
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1254 |
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1255 Raises |
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1256 ------ |
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1257 NodeNotFound |
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1258 If any of `source` is not in `G`. |
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1259 |
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1260 NetworkXUnbounded |
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1261 If the (di)graph contains a negative cost (di)cycle, the |
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1262 algorithm raises an exception to indicate the presence of the |
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1263 negative cost (di)cycle. Note: any negative weight edge in an |
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1264 undirected graph is a negative cost cycle |
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1265 """ |
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1266 for s in source: |
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1267 if s not in G: |
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1268 raise nx.NodeNotFound(f"Source {s} not in G") |
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1269 |
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1270 if pred is None: |
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1271 pred = {v: [] for v in source} |
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1272 |
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1273 if dist is None: |
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1274 dist = {v: 0 for v in source} |
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1275 |
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1276 # Heuristic Storage setup. Note: use None because nodes cannot be None |
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1277 nonexistent_edge = (None, None) |
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1278 pred_edge = {v: None for v in source} |
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1279 recent_update = {v: nonexistent_edge for v in source} |
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1280 |
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1281 G_succ = G.succ if G.is_directed() else G.adj |
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1282 inf = float("inf") |
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1283 n = len(G) |
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1284 |
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1285 count = {} |
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1286 q = deque(source) |
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1287 in_q = set(source) |
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1288 while q: |
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1289 u = q.popleft() |
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1290 in_q.remove(u) |
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1291 |
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1292 # Skip relaxations if any of the predecessors of u is in the queue. |
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1293 if all(pred_u not in in_q for pred_u in pred[u]): |
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1294 dist_u = dist[u] |
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1295 for v, e in G_succ[u].items(): |
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1296 dist_v = dist_u + weight(u, v, e) |
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1297 |
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1298 if dist_v < dist.get(v, inf): |
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1299 # In this conditional branch we are updating the path with v. |
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1300 # If it happens that some earlier update also added node v |
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1301 # that implies the existence of a negative cycle since |
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1302 # after the update node v would lie on the update path twice. |
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1303 # The update path is stored up to one of the source nodes, |
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1304 # therefore u is always in the dict recent_update |
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1305 if heuristic: |
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1306 if v in recent_update[u]: |
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1307 raise nx.NetworkXUnbounded("Negative cost cycle detected.") |
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1308 # Transfer the recent update info from u to v if the |
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1309 # same source node is the head of the update path. |
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1310 # If the source node is responsible for the cost update, |
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1311 # then clear the history and use it instead. |
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1312 if v in pred_edge and pred_edge[v] == u: |
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1313 recent_update[v] = recent_update[u] |
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1314 else: |
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1315 recent_update[v] = (u, v) |
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1316 |
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1317 if v not in in_q: |
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1318 q.append(v) |
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1319 in_q.add(v) |
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1320 count_v = count.get(v, 0) + 1 |
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1321 if count_v == n: |
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1322 raise nx.NetworkXUnbounded("Negative cost cycle detected.") |
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1323 count[v] = count_v |
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1324 dist[v] = dist_v |
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1325 pred[v] = [u] |
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1326 pred_edge[v] = u |
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1327 |
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1328 elif dist.get(v) is not None and dist_v == dist.get(v): |
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1329 pred[v].append(u) |
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1330 |
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1331 if paths is not None: |
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1332 sources = set(source) |
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1333 dsts = [target] if target is not None else pred |
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1334 for dst in dsts: |
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1335 gen = _build_paths_from_predecessors(sources, dst, pred) |
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1336 paths[dst] = next(gen) |
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1337 |
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1338 return dist |
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1339 |
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1340 |
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1341 def bellman_ford_path(G, source, target, weight="weight"): |
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1342 """Returns the shortest path from source to target in a weighted graph G. |
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1343 |
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1344 Parameters |
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1345 ---------- |
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1346 G : NetworkX graph |
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1347 |
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1348 source : node |
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1349 Starting node |
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1350 |
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1351 target : node |
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1352 Ending node |
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1353 |
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1354 weight: string, optional (default='weight') |
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1355 Edge data key corresponding to the edge weight |
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1356 |
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1357 Returns |
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1358 ------- |
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1359 path : list |
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1360 List of nodes in a shortest path. |
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1361 |
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1362 Raises |
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1363 ------ |
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1364 NodeNotFound |
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1365 If `source` is not in `G`. |
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1366 |
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1367 NetworkXNoPath |
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1368 If no path exists between source and target. |
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1369 |
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|
1370 Examples |
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|
1371 -------- |
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1372 >>> G = nx.path_graph(5) |
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1373 >>> print(nx.bellman_ford_path(G, 0, 4)) |
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1374 [0, 1, 2, 3, 4] |
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1375 |
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|
1376 Notes |
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|
1377 ----- |
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1378 Edge weight attributes must be numerical. |
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1379 Distances are calculated as sums of weighted edges traversed. |
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1380 |
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|
1381 See Also |
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|
1382 -------- |
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|
1383 dijkstra_path(), bellman_ford_path_length() |
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1384 """ |
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1385 length, path = single_source_bellman_ford(G, source, target=target, weight=weight) |
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|
1386 return path |
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1387 |
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1388 |
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1389 def bellman_ford_path_length(G, source, target, weight="weight"): |
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1390 """Returns the shortest path length from source to target |
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1391 in a weighted graph. |
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1392 |
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1393 Parameters |
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1394 ---------- |
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1395 G : NetworkX graph |
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1396 |
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|
1397 source : node label |
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|
1398 starting node for path |
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1399 |
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1400 target : node label |
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1401 ending node for path |
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1402 |
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1403 weight: string, optional (default='weight') |
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1404 Edge data key corresponding to the edge weight |
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1405 |
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1406 Returns |
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1407 ------- |
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1408 length : number |
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1409 Shortest path length. |
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1410 |
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|
1411 Raises |
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|
1412 ------ |
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1413 NodeNotFound |
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1414 If `source` is not in `G`. |
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1415 |
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|
1416 NetworkXNoPath |
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1417 If no path exists between source and target. |
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1418 |
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|
1419 Examples |
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1420 -------- |
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1421 >>> G = nx.path_graph(5) |
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1422 >>> print(nx.bellman_ford_path_length(G, 0, 4)) |
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|
1423 4 |
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|
1424 |
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|
1425 Notes |
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|
1426 ----- |
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1427 Edge weight attributes must be numerical. |
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|
1428 Distances are calculated as sums of weighted edges traversed. |
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|
1429 |
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|
1430 See Also |
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|
1431 -------- |
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|
1432 dijkstra_path_length(), bellman_ford_path() |
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|
1433 """ |
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|
1434 if source == target: |
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|
1435 return 0 |
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|
1436 |
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|
1437 weight = _weight_function(G, weight) |
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1438 |
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|
1439 length = _bellman_ford(G, [source], weight, target=target) |
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|
1440 |
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|
1441 try: |
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|
1442 return length[target] |
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|
1443 except KeyError as e: |
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|
1444 raise nx.NetworkXNoPath(f"node {target} not reachable from {source}") from e |
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|
1445 |
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|
1446 |
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|
1447 def single_source_bellman_ford_path(G, source, weight="weight"): |
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|
1448 """Compute shortest path between source and all other reachable |
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|
1449 nodes for a weighted graph. |
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|
1450 |
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|
1451 Parameters |
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|
1452 ---------- |
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|
1453 G : NetworkX graph |
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|
1454 |
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|
1455 source : node |
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|
1456 Starting node for path. |
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|
1457 |
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|
1458 weight: string, optional (default='weight') |
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|
1459 Edge data key corresponding to the edge weight |
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|
1460 |
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|
1461 Returns |
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1462 ------- |
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|
1463 paths : dictionary |
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1464 Dictionary of shortest path lengths keyed by target. |
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1465 |
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1466 Raises |
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1467 ------ |
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1468 NodeNotFound |
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1469 If `source` is not in `G`. |
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1470 |
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|
1471 Examples |
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1472 -------- |
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1473 >>> G = nx.path_graph(5) |
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1474 >>> path = nx.single_source_bellman_ford_path(G, 0) |
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|
1475 >>> path[4] |
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|
1476 [0, 1, 2, 3, 4] |
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1477 |
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1478 Notes |
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1479 ----- |
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1480 Edge weight attributes must be numerical. |
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1481 Distances are calculated as sums of weighted edges traversed. |
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1482 |
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|
1483 See Also |
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1484 -------- |
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|
1485 single_source_dijkstra(), single_source_bellman_ford() |
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1486 |
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1487 """ |
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1488 (length, path) = single_source_bellman_ford(G, source, weight=weight) |
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1489 return path |
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1490 |
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1491 |
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1492 def single_source_bellman_ford_path_length(G, source, weight="weight"): |
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1493 """Compute the shortest path length between source and all other |
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1494 reachable nodes for a weighted graph. |
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1495 |
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1496 Parameters |
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1497 ---------- |
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1498 G : NetworkX graph |
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1499 |
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1500 source : node label |
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1501 Starting node for path |
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1502 |
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1503 weight: string, optional (default='weight') |
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1504 Edge data key corresponding to the edge weight. |
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1505 |
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1506 Returns |
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1507 ------- |
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1508 length : iterator |
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1509 (target, shortest path length) iterator |
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1510 |
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1511 Raises |
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1512 ------ |
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1513 NodeNotFound |
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1514 If `source` is not in `G`. |
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1515 |
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1516 Examples |
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1517 -------- |
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1518 >>> G = nx.path_graph(5) |
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1519 >>> length = dict(nx.single_source_bellman_ford_path_length(G, 0)) |
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1520 >>> length[4] |
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1521 4 |
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1522 >>> for node in [0, 1, 2, 3, 4]: |
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1523 ... print(f"{node}: {length[node]}") |
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1524 0: 0 |
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1525 1: 1 |
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1526 2: 2 |
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1527 3: 3 |
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1528 4: 4 |
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1529 |
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1530 Notes |
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1531 ----- |
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1532 Edge weight attributes must be numerical. |
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1533 Distances are calculated as sums of weighted edges traversed. |
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1534 |
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1535 See Also |
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1536 -------- |
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1537 single_source_dijkstra(), single_source_bellman_ford() |
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1538 |
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1539 """ |
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1540 weight = _weight_function(G, weight) |
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1541 return _bellman_ford(G, [source], weight) |
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1542 |
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1543 |
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1544 def single_source_bellman_ford(G, source, target=None, weight="weight"): |
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1545 """Compute shortest paths and lengths in a weighted graph G. |
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1546 |
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1547 Uses Bellman-Ford algorithm for shortest paths. |
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1548 |
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1549 Parameters |
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1550 ---------- |
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1551 G : NetworkX graph |
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1552 |
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1553 source : node label |
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1554 Starting node for path |
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1555 |
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1556 target : node label, optional |
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1557 Ending node for path |
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1558 |
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1559 Returns |
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1560 ------- |
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1561 distance, path : pair of dictionaries, or numeric and list |
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1562 If target is None, returns a tuple of two dictionaries keyed by node. |
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1563 The first dictionary stores distance from one of the source nodes. |
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1564 The second stores the path from one of the sources to that node. |
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1565 If target is not None, returns a tuple of (distance, path) where |
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1566 distance is the distance from source to target and path is a list |
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1567 representing the path from source to target. |
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1568 |
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1569 Raises |
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1570 ------ |
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1571 NodeNotFound |
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1572 If `source` is not in `G`. |
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1573 |
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1574 Examples |
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1575 -------- |
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1576 >>> G = nx.path_graph(5) |
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1577 >>> length, path = nx.single_source_bellman_ford(G, 0) |
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1578 >>> print(length[4]) |
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1579 4 |
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1580 >>> for node in [0, 1, 2, 3, 4]: |
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1581 ... print(f"{node}: {length[node]}") |
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1582 0: 0 |
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1583 1: 1 |
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1584 2: 2 |
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1585 3: 3 |
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1586 4: 4 |
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1587 >>> path[4] |
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1588 [0, 1, 2, 3, 4] |
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1589 >>> length, path = nx.single_source_bellman_ford(G, 0, 1) |
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1590 >>> length |
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1591 1 |
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1592 >>> path |
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|
1593 [0, 1] |
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1594 |
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1595 Notes |
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1596 ----- |
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1597 Edge weight attributes must be numerical. |
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1598 Distances are calculated as sums of weighted edges traversed. |
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1599 |
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1600 See Also |
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1601 -------- |
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1602 single_source_dijkstra() |
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1603 single_source_bellman_ford_path() |
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1604 single_source_bellman_ford_path_length() |
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1605 """ |
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1606 if source == target: |
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1607 return (0, [source]) |
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1608 |
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1609 weight = _weight_function(G, weight) |
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1610 |
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1611 paths = {source: [source]} # dictionary of paths |
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1612 dist = _bellman_ford(G, [source], weight, paths=paths, target=target) |
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1613 if target is None: |
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1614 return (dist, paths) |
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1615 try: |
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1616 return (dist[target], paths[target]) |
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1617 except KeyError as e: |
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1618 msg = f"Node {target} not reachable from {source}" |
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1619 raise nx.NetworkXNoPath(msg) from e |
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1620 |
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1621 |
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1622 def all_pairs_bellman_ford_path_length(G, weight="weight"): |
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1623 """ Compute shortest path lengths between all nodes in a weighted graph. |
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1624 |
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1625 Parameters |
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1626 ---------- |
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1627 G : NetworkX graph |
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1628 |
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1629 weight: string, optional (default='weight') |
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1630 Edge data key corresponding to the edge weight |
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1631 |
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1632 Returns |
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1633 ------- |
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1634 distance : iterator |
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1635 (source, dictionary) iterator with dictionary keyed by target and |
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1636 shortest path length as the key value. |
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1637 |
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1638 Examples |
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1639 -------- |
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1640 >>> G = nx.path_graph(5) |
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1641 >>> length = dict(nx.all_pairs_bellman_ford_path_length(G)) |
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1642 >>> for node in [0, 1, 2, 3, 4]: |
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1643 ... print(f"1 - {node}: {length[1][node]}") |
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1644 1 - 0: 1 |
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1645 1 - 1: 0 |
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1646 1 - 2: 1 |
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1647 1 - 3: 2 |
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1648 1 - 4: 3 |
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1649 >>> length[3][2] |
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1650 1 |
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1651 >>> length[2][2] |
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1652 0 |
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1653 |
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1654 Notes |
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1655 ----- |
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1656 Edge weight attributes must be numerical. |
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1657 Distances are calculated as sums of weighted edges traversed. |
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1658 |
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1659 The dictionary returned only has keys for reachable node pairs. |
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1660 """ |
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1661 length = single_source_bellman_ford_path_length |
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1662 for n in G: |
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1663 yield (n, dict(length(G, n, weight=weight))) |
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1664 |
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1665 |
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1666 def all_pairs_bellman_ford_path(G, weight="weight"): |
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1667 """ Compute shortest paths between all nodes in a weighted graph. |
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1668 |
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1669 Parameters |
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1670 ---------- |
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1671 G : NetworkX graph |
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1672 |
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1673 weight: string, optional (default='weight') |
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1674 Edge data key corresponding to the edge weight |
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1675 |
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1676 Returns |
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1677 ------- |
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1678 distance : dictionary |
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1679 Dictionary, keyed by source and target, of shortest paths. |
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1680 |
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1681 Examples |
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1682 -------- |
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1683 >>> G = nx.path_graph(5) |
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1684 >>> path = dict(nx.all_pairs_bellman_ford_path(G)) |
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1685 >>> print(path[0][4]) |
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1686 [0, 1, 2, 3, 4] |
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1687 |
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1688 Notes |
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1689 ----- |
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1690 Edge weight attributes must be numerical. |
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1691 Distances are calculated as sums of weighted edges traversed. |
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1692 |
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1693 See Also |
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1694 -------- |
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1695 floyd_warshall(), all_pairs_dijkstra_path() |
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1696 |
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1697 """ |
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1698 path = single_source_bellman_ford_path |
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1699 # TODO This can be trivially parallelized. |
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1700 for n in G: |
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1701 yield (n, path(G, n, weight=weight)) |
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1702 |
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1703 |
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1704 def goldberg_radzik(G, source, weight="weight"): |
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1705 """Compute shortest path lengths and predecessors on shortest paths |
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1706 in weighted graphs. |
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1707 |
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1708 The algorithm has a running time of $O(mn)$ where $n$ is the number of |
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1709 nodes and $m$ is the number of edges. It is slower than Dijkstra but |
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1710 can handle negative edge weights. |
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1711 |
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1712 Parameters |
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1713 ---------- |
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1714 G : NetworkX graph |
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1715 The algorithm works for all types of graphs, including directed |
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1716 graphs and multigraphs. |
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1717 |
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1718 source: node label |
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1719 Starting node for path |
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1720 |
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1721 weight : string or function |
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1722 If this is a string, then edge weights will be accessed via the |
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1723 edge attribute with this key (that is, the weight of the edge |
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1724 joining `u` to `v` will be ``G.edges[u, v][weight]``). If no |
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1725 such edge attribute exists, the weight of the edge is assumed to |
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1726 be one. |
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1727 |
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1728 If this is a function, the weight of an edge is the value |
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1729 returned by the function. The function must accept exactly three |
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1730 positional arguments: the two endpoints of an edge and the |
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1731 dictionary of edge attributes for that edge. The function must |
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1732 return a number. |
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1733 |
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1734 Returns |
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1735 ------- |
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1736 pred, dist : dictionaries |
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1737 Returns two dictionaries keyed by node to predecessor in the |
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1738 path and to the distance from the source respectively. |
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1739 |
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1740 Raises |
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1741 ------ |
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1742 NodeNotFound |
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1743 If `source` is not in `G`. |
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1744 |
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1745 NetworkXUnbounded |
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1746 If the (di)graph contains a negative cost (di)cycle, the |
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1747 algorithm raises an exception to indicate the presence of the |
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1748 negative cost (di)cycle. Note: any negative weight edge in an |
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1749 undirected graph is a negative cost cycle. |
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1750 |
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1751 Examples |
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1752 -------- |
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1753 >>> G = nx.path_graph(5, create_using=nx.DiGraph()) |
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1754 >>> pred, dist = nx.goldberg_radzik(G, 0) |
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1755 >>> sorted(pred.items()) |
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1756 [(0, None), (1, 0), (2, 1), (3, 2), (4, 3)] |
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1757 >>> sorted(dist.items()) |
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1758 [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)] |
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1759 |
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1760 >>> G = nx.cycle_graph(5, create_using=nx.DiGraph()) |
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1761 >>> G[1][2]["weight"] = -7 |
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1762 >>> nx.goldberg_radzik(G, 0) |
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1763 Traceback (most recent call last): |
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1764 ... |
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1765 networkx.exception.NetworkXUnbounded: Negative cost cycle detected. |
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1766 |
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1767 Notes |
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1768 ----- |
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1769 Edge weight attributes must be numerical. |
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1770 Distances are calculated as sums of weighted edges traversed. |
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1771 |
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1772 The dictionaries returned only have keys for nodes reachable from |
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1773 the source. |
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1774 |
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1775 In the case where the (di)graph is not connected, if a component |
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1776 not containing the source contains a negative cost (di)cycle, it |
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1777 will not be detected. |
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1778 |
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1779 """ |
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1780 if source not in G: |
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1781 raise nx.NodeNotFound(f"Node {source} is not found in the graph") |
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1782 weight = _weight_function(G, weight) |
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1783 if any(weight(u, v, d) < 0 for u, v, d in nx.selfloop_edges(G, data=True)): |
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1784 raise nx.NetworkXUnbounded("Negative cost cycle detected.") |
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1785 |
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1786 if len(G) == 1: |
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1787 return {source: None}, {source: 0} |
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1788 |
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1789 if G.is_directed(): |
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1790 G_succ = G.succ |
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1791 else: |
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1792 G_succ = G.adj |
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1793 |
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1794 inf = float("inf") |
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1795 d = {u: inf for u in G} |
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1796 d[source] = 0 |
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1797 pred = {source: None} |
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1798 |
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1799 def topo_sort(relabeled): |
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1800 """Topologically sort nodes relabeled in the previous round and detect |
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1801 negative cycles. |
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1802 """ |
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1803 # List of nodes to scan in this round. Denoted by A in Goldberg and |
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1804 # Radzik's paper. |
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1805 to_scan = [] |
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1806 # In the DFS in the loop below, neg_count records for each node the |
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1807 # number of edges of negative reduced costs on the path from a DFS root |
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1808 # to the node in the DFS forest. The reduced cost of an edge (u, v) is |
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1809 # defined as d[u] + weight[u][v] - d[v]. |
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1810 # |
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1811 # neg_count also doubles as the DFS visit marker array. |
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1812 neg_count = {} |
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1813 for u in relabeled: |
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1814 # Skip visited nodes. |
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1815 if u in neg_count: |
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1816 continue |
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1817 d_u = d[u] |
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1818 # Skip nodes without out-edges of negative reduced costs. |
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1819 if all(d_u + weight(u, v, e) >= d[v] for v, e in G_succ[u].items()): |
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1820 continue |
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1821 # Nonrecursive DFS that inserts nodes reachable from u via edges of |
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1822 # nonpositive reduced costs into to_scan in (reverse) topological |
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1823 # order. |
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1824 stack = [(u, iter(G_succ[u].items()))] |
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1825 in_stack = {u} |
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1826 neg_count[u] = 0 |
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1827 while stack: |
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1828 u, it = stack[-1] |
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1829 try: |
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1830 v, e = next(it) |
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1831 except StopIteration: |
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1832 to_scan.append(u) |
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1833 stack.pop() |
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1834 in_stack.remove(u) |
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1835 continue |
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1836 t = d[u] + weight(u, v, e) |
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1837 d_v = d[v] |
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1838 if t <= d_v: |
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1839 is_neg = t < d_v |
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1840 d[v] = t |
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1841 pred[v] = u |
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1842 if v not in neg_count: |
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1843 neg_count[v] = neg_count[u] + int(is_neg) |
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1844 stack.append((v, iter(G_succ[v].items()))) |
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1845 in_stack.add(v) |
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1846 elif v in in_stack and neg_count[u] + int(is_neg) > neg_count[v]: |
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1847 # (u, v) is a back edge, and the cycle formed by the |
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1848 # path v to u and (u, v) contains at least one edge of |
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1849 # negative reduced cost. The cycle must be of negative |
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1850 # cost. |
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1851 raise nx.NetworkXUnbounded("Negative cost cycle detected.") |
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1852 to_scan.reverse() |
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1853 return to_scan |
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1854 |
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1855 def relax(to_scan): |
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1856 """Relax out-edges of relabeled nodes. |
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1857 """ |
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1858 relabeled = set() |
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1859 # Scan nodes in to_scan in topological order and relax incident |
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1860 # out-edges. Add the relabled nodes to labeled. |
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1861 for u in to_scan: |
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1862 d_u = d[u] |
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1863 for v, e in G_succ[u].items(): |
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1864 w_e = weight(u, v, e) |
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1865 if d_u + w_e < d[v]: |
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1866 d[v] = d_u + w_e |
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1867 pred[v] = u |
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1868 relabeled.add(v) |
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1869 return relabeled |
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1870 |
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1871 # Set of nodes relabled in the last round of scan operations. Denoted by B |
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1872 # in Goldberg and Radzik's paper. |
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1873 relabeled = {source} |
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1874 |
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1875 while relabeled: |
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1876 to_scan = topo_sort(relabeled) |
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1877 relabeled = relax(to_scan) |
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1878 |
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1879 d = {u: d[u] for u in pred} |
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1880 return pred, d |
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1881 |
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1882 |
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1883 def negative_edge_cycle(G, weight="weight", heuristic=True): |
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1884 """Returns True if there exists a negative edge cycle anywhere in G. |
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1885 |
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1886 Parameters |
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1887 ---------- |
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1888 G : NetworkX graph |
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1889 |
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1890 weight : string or function |
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1891 If this is a string, then edge weights will be accessed via the |
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1892 edge attribute with this key (that is, the weight of the edge |
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1893 joining `u` to `v` will be ``G.edges[u, v][weight]``). If no |
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1894 such edge attribute exists, the weight of the edge is assumed to |
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1895 be one. |
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1896 |
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1897 If this is a function, the weight of an edge is the value |
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1898 returned by the function. The function must accept exactly three |
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1899 positional arguments: the two endpoints of an edge and the |
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1900 dictionary of edge attributes for that edge. The function must |
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1901 return a number. |
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1902 |
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1903 heuristic : bool |
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1904 Determines whether to use a heuristic to early detect negative |
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1905 cycles at a negligible cost. In case of graphs with a negative cycle, |
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1906 the performance of detection increases by at least an order of magnitude. |
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1907 |
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1908 Returns |
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1909 ------- |
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1910 negative_cycle : bool |
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1911 True if a negative edge cycle exists, otherwise False. |
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1912 |
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1913 Examples |
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1914 -------- |
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1915 >>> G = nx.cycle_graph(5, create_using=nx.DiGraph()) |
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1916 >>> print(nx.negative_edge_cycle(G)) |
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1917 False |
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1918 >>> G[1][2]["weight"] = -7 |
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1919 >>> print(nx.negative_edge_cycle(G)) |
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1920 True |
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1921 |
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1922 Notes |
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1923 ----- |
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1924 Edge weight attributes must be numerical. |
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1925 Distances are calculated as sums of weighted edges traversed. |
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1926 |
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1927 This algorithm uses bellman_ford_predecessor_and_distance() but finds |
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1928 negative cycles on any component by first adding a new node connected to |
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1929 every node, and starting bellman_ford_predecessor_and_distance on that |
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1930 node. It then removes that extra node. |
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1931 """ |
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1932 newnode = generate_unique_node() |
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1933 G.add_edges_from([(newnode, n) for n in G]) |
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1934 |
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1935 try: |
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1936 bellman_ford_predecessor_and_distance(G, newnode, weight, heuristic=heuristic) |
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1937 except nx.NetworkXUnbounded: |
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1938 return True |
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1939 finally: |
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1940 G.remove_node(newnode) |
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1941 return False |
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1942 |
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1943 |
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1944 def bidirectional_dijkstra(G, source, target, weight="weight"): |
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1945 r"""Dijkstra's algorithm for shortest paths using bidirectional search. |
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1946 |
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1947 Parameters |
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1948 ---------- |
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1949 G : NetworkX graph |
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1950 |
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1951 source : node |
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1952 Starting node. |
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1953 |
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1954 target : node |
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1955 Ending node. |
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1956 |
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1957 weight : string or function |
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1958 If this is a string, then edge weights will be accessed via the |
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1959 edge attribute with this key (that is, the weight of the edge |
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1960 joining `u` to `v` will be ``G.edges[u, v][weight]``). If no |
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1961 such edge attribute exists, the weight of the edge is assumed to |
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1962 be one. |
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1963 |
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1964 If this is a function, the weight of an edge is the value |
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1965 returned by the function. The function must accept exactly three |
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1966 positional arguments: the two endpoints of an edge and the |
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1967 dictionary of edge attributes for that edge. The function must |
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1968 return a number. |
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1969 |
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1970 Returns |
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1971 ------- |
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1972 length, path : number and list |
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1973 length is the distance from source to target. |
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1974 path is a list of nodes on a path from source to target. |
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1975 |
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1976 Raises |
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1977 ------ |
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1978 NodeNotFound |
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1979 If either `source` or `target` is not in `G`. |
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1980 |
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1981 NetworkXNoPath |
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1982 If no path exists between source and target. |
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1983 |
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1984 Examples |
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1985 -------- |
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1986 >>> G = nx.path_graph(5) |
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1987 >>> length, path = nx.bidirectional_dijkstra(G, 0, 4) |
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1988 >>> print(length) |
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1989 4 |
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1990 >>> print(path) |
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1991 [0, 1, 2, 3, 4] |
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1992 |
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1993 Notes |
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1994 ----- |
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1995 Edge weight attributes must be numerical. |
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1996 Distances are calculated as sums of weighted edges traversed. |
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1997 |
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1998 In practice bidirectional Dijkstra is much more than twice as fast as |
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1999 ordinary Dijkstra. |
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2000 |
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2001 Ordinary Dijkstra expands nodes in a sphere-like manner from the |
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2002 source. The radius of this sphere will eventually be the length |
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2003 of the shortest path. Bidirectional Dijkstra will expand nodes |
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2004 from both the source and the target, making two spheres of half |
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2005 this radius. Volume of the first sphere is `\pi*r*r` while the |
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2006 others are `2*\pi*r/2*r/2`, making up half the volume. |
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2007 |
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2008 This algorithm is not guaranteed to work if edge weights |
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2009 are negative or are floating point numbers |
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2010 (overflows and roundoff errors can cause problems). |
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2011 |
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2012 See Also |
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2013 -------- |
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2014 shortest_path |
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2015 shortest_path_length |
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2016 """ |
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2017 if source not in G or target not in G: |
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2018 msg = f"Either source {source} or target {target} is not in G" |
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2019 raise nx.NodeNotFound(msg) |
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2020 |
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2021 if source == target: |
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2022 return (0, [source]) |
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2023 |
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2024 weight = _weight_function(G, weight) |
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2025 push = heappush |
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2026 pop = heappop |
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2027 # Init: [Forward, Backward] |
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2028 dists = [{}, {}] # dictionary of final distances |
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2029 paths = [{source: [source]}, {target: [target]}] # dictionary of paths |
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2030 fringe = [[], []] # heap of (distance, node) for choosing node to expand |
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2031 seen = [{source: 0}, {target: 0}] # dict of distances to seen nodes |
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2032 c = count() |
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2033 # initialize fringe heap |
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2034 push(fringe[0], (0, next(c), source)) |
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2035 push(fringe[1], (0, next(c), target)) |
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2036 # neighs for extracting correct neighbor information |
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2037 if G.is_directed(): |
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2038 neighs = [G._succ, G._pred] |
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2039 else: |
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2040 neighs = [G._adj, G._adj] |
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2041 # variables to hold shortest discovered path |
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2042 # finaldist = 1e30000 |
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2043 finalpath = [] |
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2044 dir = 1 |
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2045 while fringe[0] and fringe[1]: |
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2046 # choose direction |
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2047 # dir == 0 is forward direction and dir == 1 is back |
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2048 dir = 1 - dir |
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2049 # extract closest to expand |
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2050 (dist, _, v) = pop(fringe[dir]) |
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2051 if v in dists[dir]: |
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2052 # Shortest path to v has already been found |
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|
2053 continue |
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|
2054 # update distance |
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|
2055 dists[dir][v] = dist # equal to seen[dir][v] |
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|
2056 if v in dists[1 - dir]: |
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|
2057 # if we have scanned v in both directions we are done |
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|
2058 # we have now discovered the shortest path |
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|
2059 return (finaldist, finalpath) |
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|
2060 |
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2061 for w, d in neighs[dir][v].items(): |
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2062 if dir == 0: # forward |
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2063 vwLength = dists[dir][v] + weight(v, w, d) |
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2064 else: # back, must remember to change v,w->w,v |
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2065 vwLength = dists[dir][v] + weight(w, v, d) |
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2066 if w in dists[dir]: |
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2067 if vwLength < dists[dir][w]: |
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2068 raise ValueError("Contradictory paths found: negative weights?") |
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|
2069 elif w not in seen[dir] or vwLength < seen[dir][w]: |
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|
2070 # relaxing |
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|
2071 seen[dir][w] = vwLength |
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2072 push(fringe[dir], (vwLength, next(c), w)) |
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2073 paths[dir][w] = paths[dir][v] + [w] |
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2074 if w in seen[0] and w in seen[1]: |
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2075 # see if this path is better than than the already |
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2076 # discovered shortest path |
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2077 totaldist = seen[0][w] + seen[1][w] |
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2078 if finalpath == [] or finaldist > totaldist: |
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2079 finaldist = totaldist |
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2080 revpath = paths[1][w][:] |
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2081 revpath.reverse() |
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2082 finalpath = paths[0][w] + revpath[1:] |
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2083 raise nx.NetworkXNoPath(f"No path between {source} and {target}.") |
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2084 |
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2085 |
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2086 def johnson(G, weight="weight"): |
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2087 r"""Uses Johnson's Algorithm to compute shortest paths. |
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2088 |
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2089 Johnson's Algorithm finds a shortest path between each pair of |
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2090 nodes in a weighted graph even if negative weights are present. |
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2091 |
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|
2092 Parameters |
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2093 ---------- |
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2094 G : NetworkX graph |
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2095 |
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2096 weight : string or function |
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2097 If this is a string, then edge weights will be accessed via the |
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2098 edge attribute with this key (that is, the weight of the edge |
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2099 joining `u` to `v` will be ``G.edges[u, v][weight]``). If no |
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2100 such edge attribute exists, the weight of the edge is assumed to |
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2101 be one. |
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2102 |
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2103 If this is a function, the weight of an edge is the value |
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2104 returned by the function. The function must accept exactly three |
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2105 positional arguments: the two endpoints of an edge and the |
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2106 dictionary of edge attributes for that edge. The function must |
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2107 return a number. |
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2108 |
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2109 Returns |
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|
2110 ------- |
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2111 distance : dictionary |
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2112 Dictionary, keyed by source and target, of shortest paths. |
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2113 |
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2114 Raises |
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|
2115 ------ |
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2116 NetworkXError |
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2117 If given graph is not weighted. |
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2118 |
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|
2119 Examples |
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|
2120 -------- |
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2121 >>> graph = nx.DiGraph() |
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2122 >>> graph.add_weighted_edges_from( |
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|
2123 ... [("0", "3", 3), ("0", "1", -5), ("0", "2", 2), ("1", "2", 4), ("2", "3", 1)] |
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2124 ... ) |
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|
2125 >>> paths = nx.johnson(graph, weight="weight") |
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|
2126 >>> paths["0"]["2"] |
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|
2127 ['0', '1', '2'] |
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|
2128 |
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|
2129 Notes |
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|
2130 ----- |
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2131 Johnson's algorithm is suitable even for graphs with negative weights. It |
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|
2132 works by using the Bellman–Ford algorithm to compute a transformation of |
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|
2133 the input graph that removes all negative weights, allowing Dijkstra's |
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|
2134 algorithm to be used on the transformed graph. |
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|
2135 |
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|
2136 The time complexity of this algorithm is $O(n^2 \log n + n m)$, |
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|
2137 where $n$ is the number of nodes and $m$ the number of edges in the |
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|
2138 graph. For dense graphs, this may be faster than the Floyd–Warshall |
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|
2139 algorithm. |
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|
2140 |
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|
2141 See Also |
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2142 -------- |
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|
2143 floyd_warshall_predecessor_and_distance |
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|
2144 floyd_warshall_numpy |
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|
2145 all_pairs_shortest_path |
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|
2146 all_pairs_shortest_path_length |
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2147 all_pairs_dijkstra_path |
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2148 bellman_ford_predecessor_and_distance |
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|
2149 all_pairs_bellman_ford_path |
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2150 all_pairs_bellman_ford_path_length |
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2151 |
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2152 """ |
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2153 if not nx.is_weighted(G, weight=weight): |
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2154 raise nx.NetworkXError("Graph is not weighted.") |
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2155 |
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2156 dist = {v: 0 for v in G} |
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2157 pred = {v: [] for v in G} |
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2158 weight = _weight_function(G, weight) |
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2159 |
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2160 # Calculate distance of shortest paths |
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2161 dist_bellman = _bellman_ford(G, list(G), weight, pred=pred, dist=dist) |
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2162 |
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2163 # Update the weight function to take into account the Bellman--Ford |
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2164 # relaxation distances. |
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2165 def new_weight(u, v, d): |
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2166 return weight(u, v, d) + dist_bellman[u] - dist_bellman[v] |
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2167 |
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2168 def dist_path(v): |
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2169 paths = {v: [v]} |
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2170 _dijkstra(G, v, new_weight, paths=paths) |
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2171 return paths |
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2172 |
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2173 return {v: dist_path(v) for v in G} |