Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/flow/edmondskarp.py @ 0:4f3585e2f14b draft default tip
"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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 Edmonds-Karp algorithm for maximum flow problems. |
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3 """ |
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4 |
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5 import networkx as nx |
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6 from networkx.algorithms.flow.utils import build_residual_network |
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7 |
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8 __all__ = ["edmonds_karp"] |
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9 |
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10 |
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11 def edmonds_karp_core(R, s, t, cutoff): |
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12 """Implementation of the Edmonds-Karp algorithm. |
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13 """ |
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14 R_nodes = R.nodes |
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15 R_pred = R.pred |
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16 R_succ = R.succ |
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17 |
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18 inf = R.graph["inf"] |
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19 |
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20 def augment(path): |
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21 """Augment flow along a path from s to t. |
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22 """ |
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23 # Determine the path residual capacity. |
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24 flow = inf |
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25 it = iter(path) |
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26 u = next(it) |
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27 for v in it: |
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28 attr = R_succ[u][v] |
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29 flow = min(flow, attr["capacity"] - attr["flow"]) |
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30 u = v |
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31 if flow * 2 > inf: |
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32 raise nx.NetworkXUnbounded("Infinite capacity path, flow unbounded above.") |
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33 # Augment flow along the path. |
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34 it = iter(path) |
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35 u = next(it) |
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36 for v in it: |
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37 R_succ[u][v]["flow"] += flow |
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38 R_succ[v][u]["flow"] -= flow |
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39 u = v |
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40 return flow |
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41 |
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42 def bidirectional_bfs(): |
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43 """Bidirectional breadth-first search for an augmenting path. |
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44 """ |
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45 pred = {s: None} |
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46 q_s = [s] |
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47 succ = {t: None} |
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48 q_t = [t] |
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49 while True: |
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50 q = [] |
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51 if len(q_s) <= len(q_t): |
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52 for u in q_s: |
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53 for v, attr in R_succ[u].items(): |
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54 if v not in pred and attr["flow"] < attr["capacity"]: |
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55 pred[v] = u |
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56 if v in succ: |
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57 return v, pred, succ |
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58 q.append(v) |
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59 if not q: |
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60 return None, None, None |
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61 q_s = q |
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62 else: |
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63 for u in q_t: |
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64 for v, attr in R_pred[u].items(): |
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65 if v not in succ and attr["flow"] < attr["capacity"]: |
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66 succ[v] = u |
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67 if v in pred: |
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68 return v, pred, succ |
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69 q.append(v) |
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70 if not q: |
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71 return None, None, None |
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72 q_t = q |
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73 |
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74 # Look for shortest augmenting paths using breadth-first search. |
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75 flow_value = 0 |
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76 while flow_value < cutoff: |
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77 v, pred, succ = bidirectional_bfs() |
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78 if pred is None: |
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79 break |
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80 path = [v] |
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81 # Trace a path from s to v. |
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82 u = v |
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83 while u != s: |
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84 u = pred[u] |
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85 path.append(u) |
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86 path.reverse() |
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87 # Trace a path from v to t. |
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88 u = v |
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89 while u != t: |
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90 u = succ[u] |
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91 path.append(u) |
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92 flow_value += augment(path) |
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93 |
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94 return flow_value |
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95 |
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96 |
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97 def edmonds_karp_impl(G, s, t, capacity, residual, cutoff): |
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98 """Implementation of the Edmonds-Karp algorithm. |
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99 """ |
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100 if s not in G: |
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101 raise nx.NetworkXError(f"node {str(s)} not in graph") |
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102 if t not in G: |
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103 raise nx.NetworkXError(f"node {str(t)} not in graph") |
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104 if s == t: |
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105 raise nx.NetworkXError("source and sink are the same node") |
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106 |
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107 if residual is None: |
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108 R = build_residual_network(G, capacity) |
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109 else: |
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110 R = residual |
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111 |
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112 # Initialize/reset the residual network. |
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113 for u in R: |
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114 for e in R[u].values(): |
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115 e["flow"] = 0 |
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116 |
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117 if cutoff is None: |
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118 cutoff = float("inf") |
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119 R.graph["flow_value"] = edmonds_karp_core(R, s, t, cutoff) |
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120 |
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121 return R |
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122 |
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123 |
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124 def edmonds_karp( |
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125 G, s, t, capacity="capacity", residual=None, value_only=False, cutoff=None |
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126 ): |
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127 """Find a maximum single-commodity flow using the Edmonds-Karp algorithm. |
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128 |
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129 This function returns the residual network resulting after computing |
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130 the maximum flow. See below for details about the conventions |
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131 NetworkX uses for defining residual networks. |
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132 |
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133 This algorithm has a running time of $O(n m^2)$ for $n$ nodes and $m$ |
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134 edges. |
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135 |
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136 |
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137 Parameters |
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138 ---------- |
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139 G : NetworkX graph |
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140 Edges of the graph are expected to have an attribute called |
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141 'capacity'. If this attribute is not present, the edge is |
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142 considered to have infinite capacity. |
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143 |
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144 s : node |
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145 Source node for the flow. |
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146 |
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147 t : node |
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148 Sink node for the flow. |
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149 |
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150 capacity : string |
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151 Edges of the graph G are expected to have an attribute capacity |
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152 that indicates how much flow the edge can support. If this |
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153 attribute is not present, the edge is considered to have |
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154 infinite capacity. Default value: 'capacity'. |
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155 |
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156 residual : NetworkX graph |
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157 Residual network on which the algorithm is to be executed. If None, a |
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158 new residual network is created. Default value: None. |
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159 |
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160 value_only : bool |
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161 If True compute only the value of the maximum flow. This parameter |
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162 will be ignored by this algorithm because it is not applicable. |
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163 |
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164 cutoff : integer, float |
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165 If specified, the algorithm will terminate when the flow value reaches |
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166 or exceeds the cutoff. In this case, it may be unable to immediately |
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167 determine a minimum cut. Default value: None. |
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168 |
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169 Returns |
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170 ------- |
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171 R : NetworkX DiGraph |
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172 Residual network after computing the maximum flow. |
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173 |
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174 Raises |
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175 ------ |
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176 NetworkXError |
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177 The algorithm does not support MultiGraph and MultiDiGraph. If |
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178 the input graph is an instance of one of these two classes, a |
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179 NetworkXError is raised. |
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180 |
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181 NetworkXUnbounded |
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182 If the graph has a path of infinite capacity, the value of a |
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183 feasible flow on the graph is unbounded above and the function |
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184 raises a NetworkXUnbounded. |
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185 |
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186 See also |
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187 -------- |
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188 :meth:`maximum_flow` |
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189 :meth:`minimum_cut` |
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190 :meth:`preflow_push` |
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191 :meth:`shortest_augmenting_path` |
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192 |
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193 Notes |
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194 ----- |
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195 The residual network :samp:`R` from an input graph :samp:`G` has the |
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196 same nodes as :samp:`G`. :samp:`R` is a DiGraph that contains a pair |
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197 of edges :samp:`(u, v)` and :samp:`(v, u)` iff :samp:`(u, v)` is not a |
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198 self-loop, and at least one of :samp:`(u, v)` and :samp:`(v, u)` exists |
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199 in :samp:`G`. |
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200 |
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201 For each edge :samp:`(u, v)` in :samp:`R`, :samp:`R[u][v]['capacity']` |
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202 is equal to the capacity of :samp:`(u, v)` in :samp:`G` if it exists |
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203 in :samp:`G` or zero otherwise. If the capacity is infinite, |
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204 :samp:`R[u][v]['capacity']` will have a high arbitrary finite value |
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205 that does not affect the solution of the problem. This value is stored in |
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206 :samp:`R.graph['inf']`. For each edge :samp:`(u, v)` in :samp:`R`, |
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207 :samp:`R[u][v]['flow']` represents the flow function of :samp:`(u, v)` and |
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208 satisfies :samp:`R[u][v]['flow'] == -R[v][u]['flow']`. |
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209 |
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210 The flow value, defined as the total flow into :samp:`t`, the sink, is |
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211 stored in :samp:`R.graph['flow_value']`. If :samp:`cutoff` is not |
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212 specified, reachability to :samp:`t` using only edges :samp:`(u, v)` such |
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213 that :samp:`R[u][v]['flow'] < R[u][v]['capacity']` induces a minimum |
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214 :samp:`s`-:samp:`t` cut. |
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215 |
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216 Examples |
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217 -------- |
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218 >>> from networkx.algorithms.flow import edmonds_karp |
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219 |
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220 The functions that implement flow algorithms and output a residual |
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221 network, such as this one, are not imported to the base NetworkX |
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222 namespace, so you have to explicitly import them from the flow package. |
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223 |
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224 >>> G = nx.DiGraph() |
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225 >>> G.add_edge("x", "a", capacity=3.0) |
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226 >>> G.add_edge("x", "b", capacity=1.0) |
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227 >>> G.add_edge("a", "c", capacity=3.0) |
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228 >>> G.add_edge("b", "c", capacity=5.0) |
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229 >>> G.add_edge("b", "d", capacity=4.0) |
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230 >>> G.add_edge("d", "e", capacity=2.0) |
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231 >>> G.add_edge("c", "y", capacity=2.0) |
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232 >>> G.add_edge("e", "y", capacity=3.0) |
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233 >>> R = edmonds_karp(G, "x", "y") |
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234 >>> flow_value = nx.maximum_flow_value(G, "x", "y") |
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235 >>> flow_value |
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236 3.0 |
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237 >>> flow_value == R.graph["flow_value"] |
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238 True |
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239 |
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240 """ |
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241 R = edmonds_karp_impl(G, s, t, capacity, residual, cutoff) |
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242 R.graph["algorithm"] = "edmonds_karp" |
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243 return R |