Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/shortest_paths/dense.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 """Floyd-Warshall algorithm for shortest paths. |
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2 """ |
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3 import networkx as nx |
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4 |
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5 __all__ = [ |
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6 "floyd_warshall", |
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7 "floyd_warshall_predecessor_and_distance", |
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8 "reconstruct_path", |
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9 "floyd_warshall_numpy", |
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10 ] |
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11 |
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12 |
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13 def floyd_warshall_numpy(G, nodelist=None, weight="weight"): |
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14 """Find all-pairs shortest path lengths using Floyd's algorithm. |
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15 |
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16 Parameters |
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17 ---------- |
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18 G : NetworkX graph |
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19 |
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20 nodelist : list, optional |
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21 The rows and columns are ordered by the nodes in nodelist. |
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22 If nodelist is None then the ordering is produced by G.nodes(). |
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23 |
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24 weight: string, optional (default= 'weight') |
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25 Edge data key corresponding to the edge weight. |
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26 |
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27 Returns |
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28 ------- |
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29 distance : NumPy matrix |
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30 A matrix of shortest path distances between nodes. |
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31 If there is no path between to nodes the corresponding matrix entry |
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32 will be Inf. |
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33 |
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34 Notes |
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35 ------ |
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36 Floyd's algorithm is appropriate for finding shortest paths in |
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37 dense graphs or graphs with negative weights when Dijkstra's |
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38 algorithm fails. This algorithm can still fail if there are negative |
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39 cycles. It has running time $O(n^3)$ with running space of $O(n^2)$. |
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40 """ |
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41 try: |
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42 import numpy as np |
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43 except ImportError as e: |
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44 raise ImportError("to_numpy_array() requires numpy: http://numpy.org/ ") from e |
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45 |
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46 # To handle cases when an edge has weight=0, we must make sure that |
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47 # nonedges are not given the value 0 as well. |
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48 A = nx.to_numpy_array( |
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49 G, nodelist=nodelist, multigraph_weight=min, weight=weight, nonedge=np.inf |
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50 ) |
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51 n, m = A.shape |
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52 np.fill_diagonal(A, 0) # diagonal elements should be zero |
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53 for i in range(n): |
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54 # The second term has the same shape as A due to broadcasting |
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55 A = np.minimum(A, A[i, :][np.newaxis, :] + A[:, i][:, np.newaxis]) |
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56 return A |
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57 |
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58 |
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59 def floyd_warshall_predecessor_and_distance(G, weight="weight"): |
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60 """Find all-pairs shortest path lengths using Floyd's algorithm. |
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61 |
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62 Parameters |
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63 ---------- |
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64 G : NetworkX graph |
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65 |
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66 weight: string, optional (default= 'weight') |
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67 Edge data key corresponding to the edge weight. |
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68 |
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69 Returns |
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70 ------- |
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71 predecessor,distance : dictionaries |
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72 Dictionaries, keyed by source and target, of predecessors and distances |
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73 in the shortest path. |
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74 |
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75 Examples |
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76 -------- |
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77 >>> G = nx.DiGraph() |
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78 >>> G.add_weighted_edges_from( |
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79 ... [ |
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80 ... ("s", "u", 10), |
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81 ... ("s", "x", 5), |
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82 ... ("u", "v", 1), |
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83 ... ("u", "x", 2), |
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84 ... ("v", "y", 1), |
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85 ... ("x", "u", 3), |
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86 ... ("x", "v", 5), |
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87 ... ("x", "y", 2), |
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88 ... ("y", "s", 7), |
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89 ... ("y", "v", 6), |
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90 ... ] |
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91 ... ) |
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92 >>> predecessors, _ = nx.floyd_warshall_predecessor_and_distance(G) |
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93 >>> print(nx.reconstruct_path("s", "v", predecessors)) |
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94 ['s', 'x', 'u', 'v'] |
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95 |
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96 Notes |
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97 ------ |
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98 Floyd's algorithm is appropriate for finding shortest paths |
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99 in dense graphs or graphs with negative weights when Dijkstra's algorithm |
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100 fails. This algorithm can still fail if there are negative cycles. |
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101 It has running time $O(n^3)$ with running space of $O(n^2)$. |
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102 |
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103 See Also |
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104 -------- |
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105 floyd_warshall |
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106 floyd_warshall_numpy |
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107 all_pairs_shortest_path |
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108 all_pairs_shortest_path_length |
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109 """ |
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110 from collections import defaultdict |
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111 |
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112 # dictionary-of-dictionaries representation for dist and pred |
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113 # use some defaultdict magick here |
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114 # for dist the default is the floating point inf value |
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115 dist = defaultdict(lambda: defaultdict(lambda: float("inf"))) |
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116 for u in G: |
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117 dist[u][u] = 0 |
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118 pred = defaultdict(dict) |
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119 # initialize path distance dictionary to be the adjacency matrix |
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120 # also set the distance to self to 0 (zero diagonal) |
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121 undirected = not G.is_directed() |
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122 for u, v, d in G.edges(data=True): |
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123 e_weight = d.get(weight, 1.0) |
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124 dist[u][v] = min(e_weight, dist[u][v]) |
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125 pred[u][v] = u |
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126 if undirected: |
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127 dist[v][u] = min(e_weight, dist[v][u]) |
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128 pred[v][u] = v |
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129 for w in G: |
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130 dist_w = dist[w] # save recomputation |
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131 for u in G: |
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132 dist_u = dist[u] # save recomputation |
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133 for v in G: |
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134 d = dist_u[w] + dist_w[v] |
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135 if dist_u[v] > d: |
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136 dist_u[v] = d |
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137 pred[u][v] = pred[w][v] |
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138 return dict(pred), dict(dist) |
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139 |
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140 |
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141 def reconstruct_path(source, target, predecessors): |
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142 """Reconstruct a path from source to target using the predecessors |
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143 dict as returned by floyd_warshall_predecessor_and_distance |
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144 |
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145 Parameters |
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146 ---------- |
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147 source : node |
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148 Starting node for path |
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149 |
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150 target : node |
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151 Ending node for path |
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152 |
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153 predecessors: dictionary |
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154 Dictionary, keyed by source and target, of predecessors in the |
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155 shortest path, as returned by floyd_warshall_predecessor_and_distance |
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156 |
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157 Returns |
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158 ------- |
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159 path : list |
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160 A list of nodes containing the shortest path from source to target |
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161 |
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162 If source and target are the same, an empty list is returned |
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163 |
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164 Notes |
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165 ------ |
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166 This function is meant to give more applicability to the |
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167 floyd_warshall_predecessor_and_distance function |
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168 |
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169 See Also |
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170 -------- |
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171 floyd_warshall_predecessor_and_distance |
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172 """ |
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173 if source == target: |
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174 return [] |
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175 prev = predecessors[source] |
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176 curr = prev[target] |
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177 path = [target, curr] |
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178 while curr != source: |
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179 curr = prev[curr] |
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180 path.append(curr) |
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181 return list(reversed(path)) |
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182 |
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183 |
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184 def floyd_warshall(G, weight="weight"): |
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185 """Find all-pairs shortest path lengths using Floyd's algorithm. |
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186 |
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187 Parameters |
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188 ---------- |
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189 G : NetworkX graph |
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190 |
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191 weight: string, optional (default= 'weight') |
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192 Edge data key corresponding to the edge weight. |
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193 |
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194 |
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195 Returns |
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196 ------- |
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197 distance : dict |
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198 A dictionary, keyed by source and target, of shortest paths distances |
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199 between nodes. |
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200 |
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201 Notes |
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202 ------ |
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203 Floyd's algorithm is appropriate for finding shortest paths |
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204 in dense graphs or graphs with negative weights when Dijkstra's algorithm |
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205 fails. This algorithm can still fail if there are negative cycles. |
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206 It has running time $O(n^3)$ with running space of $O(n^2)$. |
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207 |
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208 See Also |
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209 -------- |
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210 floyd_warshall_predecessor_and_distance |
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211 floyd_warshall_numpy |
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212 all_pairs_shortest_path |
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213 all_pairs_shortest_path_length |
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214 """ |
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215 # could make this its own function to reduce memory costs |
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216 return floyd_warshall_predecessor_and_distance(G, weight=weight)[1] |