Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/flow/capacityscaling.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 Capacity scaling minimum cost flow algorithm. |
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3 """ |
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4 |
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5 __all__ = ["capacity_scaling"] |
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6 |
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7 from itertools import chain |
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8 from math import log |
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9 import networkx as nx |
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10 from ...utils import BinaryHeap |
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11 from ...utils import generate_unique_node |
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12 from ...utils import not_implemented_for |
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13 from ...utils import arbitrary_element |
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14 |
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15 |
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16 def _detect_unboundedness(R): |
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17 """Detect infinite-capacity negative cycles. |
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18 """ |
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19 s = generate_unique_node() |
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20 G = nx.DiGraph() |
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21 G.add_nodes_from(R) |
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22 |
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23 # Value simulating infinity. |
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24 inf = R.graph["inf"] |
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25 # True infinity. |
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26 f_inf = float("inf") |
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27 for u in R: |
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28 for v, e in R[u].items(): |
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29 # Compute the minimum weight of infinite-capacity (u, v) edges. |
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30 w = f_inf |
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31 for k, e in e.items(): |
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32 if e["capacity"] == inf: |
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33 w = min(w, e["weight"]) |
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34 if w != f_inf: |
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35 G.add_edge(u, v, weight=w) |
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36 |
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37 if nx.negative_edge_cycle(G): |
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38 raise nx.NetworkXUnbounded( |
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39 "Negative cost cycle of infinite capacity found. " |
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40 "Min cost flow may be unbounded below." |
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41 ) |
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42 |
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43 |
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44 @not_implemented_for("undirected") |
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45 def _build_residual_network(G, demand, capacity, weight): |
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46 """Build a residual network and initialize a zero flow. |
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47 """ |
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48 if sum(G.nodes[u].get(demand, 0) for u in G) != 0: |
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49 raise nx.NetworkXUnfeasible("Sum of the demands should be 0.") |
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50 |
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51 R = nx.MultiDiGraph() |
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52 R.add_nodes_from( |
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53 (u, {"excess": -G.nodes[u].get(demand, 0), "potential": 0}) for u in G |
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54 ) |
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55 |
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56 inf = float("inf") |
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57 # Detect selfloops with infinite capacities and negative weights. |
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58 for u, v, e in nx.selfloop_edges(G, data=True): |
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59 if e.get(weight, 0) < 0 and e.get(capacity, inf) == inf: |
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60 raise nx.NetworkXUnbounded( |
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61 "Negative cost cycle of infinite capacity found. " |
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62 "Min cost flow may be unbounded below." |
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63 ) |
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64 |
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65 # Extract edges with positive capacities. Self loops excluded. |
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66 if G.is_multigraph(): |
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67 edge_list = [ |
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68 (u, v, k, e) |
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69 for u, v, k, e in G.edges(data=True, keys=True) |
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70 if u != v and e.get(capacity, inf) > 0 |
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71 ] |
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72 else: |
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73 edge_list = [ |
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74 (u, v, 0, e) |
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75 for u, v, e in G.edges(data=True) |
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76 if u != v and e.get(capacity, inf) > 0 |
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77 ] |
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78 # Simulate infinity with the larger of the sum of absolute node imbalances |
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79 # the sum of finite edge capacities or any positive value if both sums are |
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80 # zero. This allows the infinite-capacity edges to be distinguished for |
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81 # unboundedness detection and directly participate in residual capacity |
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82 # calculation. |
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83 inf = ( |
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84 max( |
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85 sum(abs(R.nodes[u]["excess"]) for u in R), |
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86 2 |
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87 * sum( |
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88 e[capacity] |
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89 for u, v, k, e in edge_list |
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90 if capacity in e and e[capacity] != inf |
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91 ), |
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92 ) |
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93 or 1 |
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94 ) |
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95 for u, v, k, e in edge_list: |
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96 r = min(e.get(capacity, inf), inf) |
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97 w = e.get(weight, 0) |
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98 # Add both (u, v) and (v, u) into the residual network marked with the |
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99 # original key. (key[1] == True) indicates the (u, v) is in the |
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100 # original network. |
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101 R.add_edge(u, v, key=(k, True), capacity=r, weight=w, flow=0) |
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102 R.add_edge(v, u, key=(k, False), capacity=0, weight=-w, flow=0) |
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103 |
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104 # Record the value simulating infinity. |
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105 R.graph["inf"] = inf |
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106 |
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107 _detect_unboundedness(R) |
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108 |
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109 return R |
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110 |
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111 |
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112 def _build_flow_dict(G, R, capacity, weight): |
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113 """Build a flow dictionary from a residual network. |
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114 """ |
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115 inf = float("inf") |
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116 flow_dict = {} |
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117 if G.is_multigraph(): |
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118 for u in G: |
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119 flow_dict[u] = {} |
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120 for v, es in G[u].items(): |
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121 flow_dict[u][v] = { |
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122 # Always saturate negative selfloops. |
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123 k: ( |
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124 0 |
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125 if ( |
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126 u != v or e.get(capacity, inf) <= 0 or e.get(weight, 0) >= 0 |
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127 ) |
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128 else e[capacity] |
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129 ) |
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130 for k, e in es.items() |
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131 } |
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132 for v, es in R[u].items(): |
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133 if v in flow_dict[u]: |
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134 flow_dict[u][v].update( |
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135 (k[0], e["flow"]) for k, e in es.items() if e["flow"] > 0 |
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136 ) |
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137 else: |
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138 for u in G: |
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139 flow_dict[u] = { |
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140 # Always saturate negative selfloops. |
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141 v: ( |
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142 0 |
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143 if (u != v or e.get(capacity, inf) <= 0 or e.get(weight, 0) >= 0) |
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144 else e[capacity] |
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145 ) |
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146 for v, e in G[u].items() |
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147 } |
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148 flow_dict[u].update( |
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149 (v, e["flow"]) |
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150 for v, es in R[u].items() |
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151 for e in es.values() |
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152 if e["flow"] > 0 |
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153 ) |
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154 return flow_dict |
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155 |
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156 |
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157 def capacity_scaling( |
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158 G, demand="demand", capacity="capacity", weight="weight", heap=BinaryHeap |
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159 ): |
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160 r"""Find a minimum cost flow satisfying all demands in digraph G. |
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161 |
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162 This is a capacity scaling successive shortest augmenting path algorithm. |
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163 |
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164 G is a digraph with edge costs and capacities and in which nodes |
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165 have demand, i.e., they want to send or receive some amount of |
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166 flow. A negative demand means that the node wants to send flow, a |
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167 positive demand means that the node want to receive flow. A flow on |
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168 the digraph G satisfies all demand if the net flow into each node |
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169 is equal to the demand of that node. |
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170 |
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171 Parameters |
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172 ---------- |
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173 G : NetworkX graph |
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174 DiGraph or MultiDiGraph on which a minimum cost flow satisfying all |
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175 demands is to be found. |
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176 |
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177 demand : string |
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178 Nodes of the graph G are expected to have an attribute demand |
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179 that indicates how much flow a node wants to send (negative |
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180 demand) or receive (positive demand). Note that the sum of the |
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181 demands should be 0 otherwise the problem in not feasible. If |
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182 this attribute is not present, a node is considered to have 0 |
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183 demand. Default value: 'demand'. |
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184 |
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185 capacity : string |
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186 Edges of the graph G are expected to have an attribute capacity |
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187 that indicates how much flow the edge can support. If this |
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188 attribute is not present, the edge is considered to have |
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189 infinite capacity. Default value: 'capacity'. |
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190 |
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191 weight : string |
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192 Edges of the graph G are expected to have an attribute weight |
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193 that indicates the cost incurred by sending one unit of flow on |
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194 that edge. If not present, the weight is considered to be 0. |
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195 Default value: 'weight'. |
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196 |
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197 heap : class |
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198 Type of heap to be used in the algorithm. It should be a subclass of |
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199 :class:`MinHeap` or implement a compatible interface. |
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200 |
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201 If a stock heap implementation is to be used, :class:`BinaryHeap` is |
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202 recommended over :class:`PairingHeap` for Python implementations without |
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203 optimized attribute accesses (e.g., CPython) despite a slower |
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204 asymptotic running time. For Python implementations with optimized |
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205 attribute accesses (e.g., PyPy), :class:`PairingHeap` provides better |
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206 performance. Default value: :class:`BinaryHeap`. |
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207 |
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208 Returns |
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209 ------- |
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210 flowCost : integer |
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211 Cost of a minimum cost flow satisfying all demands. |
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212 |
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213 flowDict : dictionary |
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214 If G is a digraph, a dict-of-dicts keyed by nodes such that |
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215 flowDict[u][v] is the flow on edge (u, v). |
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216 If G is a MultiDiGraph, a dict-of-dicts-of-dicts keyed by nodes |
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217 so that flowDict[u][v][key] is the flow on edge (u, v, key). |
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218 |
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219 Raises |
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220 ------ |
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221 NetworkXError |
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222 This exception is raised if the input graph is not directed, |
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223 not connected. |
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224 |
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225 NetworkXUnfeasible |
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226 This exception is raised in the following situations: |
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227 |
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228 * The sum of the demands is not zero. Then, there is no |
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229 flow satisfying all demands. |
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230 * There is no flow satisfying all demand. |
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231 |
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232 NetworkXUnbounded |
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233 This exception is raised if the digraph G has a cycle of |
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234 negative cost and infinite capacity. Then, the cost of a flow |
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235 satisfying all demands is unbounded below. |
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236 |
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237 Notes |
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238 ----- |
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239 This algorithm does not work if edge weights are floating-point numbers. |
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240 |
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241 See also |
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242 -------- |
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243 :meth:`network_simplex` |
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244 |
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245 Examples |
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246 -------- |
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247 A simple example of a min cost flow problem. |
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248 |
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249 >>> G = nx.DiGraph() |
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250 >>> G.add_node("a", demand=-5) |
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251 >>> G.add_node("d", demand=5) |
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252 >>> G.add_edge("a", "b", weight=3, capacity=4) |
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253 >>> G.add_edge("a", "c", weight=6, capacity=10) |
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254 >>> G.add_edge("b", "d", weight=1, capacity=9) |
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255 >>> G.add_edge("c", "d", weight=2, capacity=5) |
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256 >>> flowCost, flowDict = nx.capacity_scaling(G) |
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257 >>> flowCost |
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258 24 |
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259 >>> flowDict # doctest: +SKIP |
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260 {'a': {'c': 1, 'b': 4}, 'c': {'d': 1}, 'b': {'d': 4}, 'd': {}} |
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261 |
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262 It is possible to change the name of the attributes used for the |
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263 algorithm. |
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264 |
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265 >>> G = nx.DiGraph() |
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266 >>> G.add_node("p", spam=-4) |
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267 >>> G.add_node("q", spam=2) |
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268 >>> G.add_node("a", spam=-2) |
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269 >>> G.add_node("d", spam=-1) |
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270 >>> G.add_node("t", spam=2) |
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271 >>> G.add_node("w", spam=3) |
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272 >>> G.add_edge("p", "q", cost=7, vacancies=5) |
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273 >>> G.add_edge("p", "a", cost=1, vacancies=4) |
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274 >>> G.add_edge("q", "d", cost=2, vacancies=3) |
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275 >>> G.add_edge("t", "q", cost=1, vacancies=2) |
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276 >>> G.add_edge("a", "t", cost=2, vacancies=4) |
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277 >>> G.add_edge("d", "w", cost=3, vacancies=4) |
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278 >>> G.add_edge("t", "w", cost=4, vacancies=1) |
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279 >>> flowCost, flowDict = nx.capacity_scaling( |
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280 ... G, demand="spam", capacity="vacancies", weight="cost" |
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281 ... ) |
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282 >>> flowCost |
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283 37 |
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284 >>> flowDict # doctest: +SKIP |
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285 {'a': {'t': 4}, 'd': {'w': 2}, 'q': {'d': 1}, 'p': {'q': 2, 'a': 2}, 't': {'q': 1, 'w': 1}, 'w': {}} |
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286 """ |
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287 R = _build_residual_network(G, demand, capacity, weight) |
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288 |
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289 inf = float("inf") |
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290 # Account cost of negative selfloops. |
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291 flow_cost = sum( |
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292 0 |
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293 if e.get(capacity, inf) <= 0 or e.get(weight, 0) >= 0 |
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294 else e[capacity] * e[weight] |
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295 for u, v, e in nx.selfloop_edges(G, data=True) |
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296 ) |
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297 |
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298 # Determine the maxmimum edge capacity. |
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299 wmax = max(chain([-inf], (e["capacity"] for u, v, e in R.edges(data=True)))) |
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300 if wmax == -inf: |
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301 # Residual network has no edges. |
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302 return flow_cost, _build_flow_dict(G, R, capacity, weight) |
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303 |
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304 R_nodes = R.nodes |
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305 R_succ = R.succ |
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306 |
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307 delta = 2 ** int(log(wmax, 2)) |
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308 while delta >= 1: |
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309 # Saturate Δ-residual edges with negative reduced costs to achieve |
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310 # Δ-optimality. |
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311 for u in R: |
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312 p_u = R_nodes[u]["potential"] |
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313 for v, es in R_succ[u].items(): |
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314 for k, e in es.items(): |
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315 flow = e["capacity"] - e["flow"] |
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316 if e["weight"] - p_u + R_nodes[v]["potential"] < 0: |
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317 flow = e["capacity"] - e["flow"] |
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318 if flow >= delta: |
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319 e["flow"] += flow |
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320 R_succ[v][u][(k[0], not k[1])]["flow"] -= flow |
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321 R_nodes[u]["excess"] -= flow |
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322 R_nodes[v]["excess"] += flow |
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323 # Determine the Δ-active nodes. |
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324 S = set() |
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325 T = set() |
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326 S_add = S.add |
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327 S_remove = S.remove |
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328 T_add = T.add |
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329 T_remove = T.remove |
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330 for u in R: |
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331 excess = R_nodes[u]["excess"] |
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332 if excess >= delta: |
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333 S_add(u) |
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334 elif excess <= -delta: |
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335 T_add(u) |
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336 # Repeatedly augment flow from S to T along shortest paths until |
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337 # Δ-feasibility is achieved. |
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338 while S and T: |
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339 s = arbitrary_element(S) |
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340 t = None |
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341 # Search for a shortest path in terms of reduce costs from s to |
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342 # any t in T in the Δ-residual network. |
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343 d = {} |
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344 pred = {s: None} |
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345 h = heap() |
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346 h_insert = h.insert |
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347 h_get = h.get |
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348 h_insert(s, 0) |
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349 while h: |
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350 u, d_u = h.pop() |
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351 d[u] = d_u |
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352 if u in T: |
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353 # Path found. |
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354 t = u |
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355 break |
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356 p_u = R_nodes[u]["potential"] |
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357 for v, es in R_succ[u].items(): |
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358 if v in d: |
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359 continue |
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360 wmin = inf |
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361 # Find the minimum-weighted (u, v) Δ-residual edge. |
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362 for k, e in es.items(): |
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363 if e["capacity"] - e["flow"] >= delta: |
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364 w = e["weight"] |
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365 if w < wmin: |
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366 wmin = w |
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367 kmin = k |
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368 emin = e |
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369 if wmin == inf: |
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370 continue |
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371 # Update the distance label of v. |
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372 d_v = d_u + wmin - p_u + R_nodes[v]["potential"] |
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373 if h_insert(v, d_v): |
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374 pred[v] = (u, kmin, emin) |
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375 if t is not None: |
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376 # Augment Δ units of flow from s to t. |
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377 while u != s: |
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378 v = u |
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379 u, k, e = pred[v] |
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380 e["flow"] += delta |
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381 R_succ[v][u][(k[0], not k[1])]["flow"] -= delta |
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382 # Account node excess and deficit. |
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383 R_nodes[s]["excess"] -= delta |
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384 R_nodes[t]["excess"] += delta |
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385 if R_nodes[s]["excess"] < delta: |
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386 S_remove(s) |
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387 if R_nodes[t]["excess"] > -delta: |
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388 T_remove(t) |
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389 # Update node potentials. |
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390 d_t = d[t] |
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391 for u, d_u in d.items(): |
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392 R_nodes[u]["potential"] -= d_u - d_t |
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393 else: |
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394 # Path not found. |
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395 S_remove(s) |
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396 delta //= 2 |
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397 |
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398 if any(R.nodes[u]["excess"] != 0 for u in R): |
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399 raise nx.NetworkXUnfeasible("No flow satisfying all demands.") |
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400 |
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401 # Calculate the flow cost. |
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402 for u in R: |
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403 for v, es in R_succ[u].items(): |
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404 for e in es.values(): |
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405 flow = e["flow"] |
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406 if flow > 0: |
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407 flow_cost += flow * e["weight"] |
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408 |
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409 return flow_cost, _build_flow_dict(G, R, capacity, weight) |