Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/flow/networksimplex.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 Minimum cost flow algorithms on directed connected graphs. |
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3 """ |
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4 |
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5 __all__ = ["network_simplex"] |
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6 |
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7 from itertools import chain, islice, repeat |
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8 from math import ceil, sqrt |
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9 import networkx as nx |
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10 from networkx.utils import not_implemented_for |
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11 |
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12 |
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13 @not_implemented_for("undirected") |
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14 def network_simplex(G, demand="demand", capacity="capacity", weight="weight"): |
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15 r"""Find a minimum cost flow satisfying all demands in digraph G. |
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16 |
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17 This is a primal network simplex algorithm that uses the leaving |
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18 arc rule to prevent cycling. |
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19 |
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20 G is a digraph with edge costs and capacities and in which nodes |
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21 have demand, i.e., they want to send or receive some amount of |
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22 flow. A negative demand means that the node wants to send flow, a |
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23 positive demand means that the node want to receive flow. A flow on |
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24 the digraph G satisfies all demand if the net flow into each node |
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25 is equal to the demand of that node. |
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26 |
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27 Parameters |
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28 ---------- |
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29 G : NetworkX graph |
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30 DiGraph on which a minimum cost flow satisfying all demands is |
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31 to be found. |
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32 |
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33 demand : string |
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34 Nodes of the graph G are expected to have an attribute demand |
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35 that indicates how much flow a node wants to send (negative |
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36 demand) or receive (positive demand). Note that the sum of the |
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37 demands should be 0 otherwise the problem in not feasible. If |
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38 this attribute is not present, a node is considered to have 0 |
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39 demand. Default value: 'demand'. |
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40 |
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41 capacity : string |
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42 Edges of the graph G are expected to have an attribute capacity |
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43 that indicates how much flow the edge can support. If this |
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44 attribute is not present, the edge is considered to have |
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45 infinite capacity. Default value: 'capacity'. |
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46 |
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47 weight : string |
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48 Edges of the graph G are expected to have an attribute weight |
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49 that indicates the cost incurred by sending one unit of flow on |
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50 that edge. If not present, the weight is considered to be 0. |
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51 Default value: 'weight'. |
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52 |
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53 Returns |
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54 ------- |
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55 flowCost : integer, float |
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56 Cost of a minimum cost flow satisfying all demands. |
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57 |
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58 flowDict : dictionary |
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59 Dictionary of dictionaries keyed by nodes such that |
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60 flowDict[u][v] is the flow edge (u, v). |
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61 |
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62 Raises |
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63 ------ |
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64 NetworkXError |
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65 This exception is raised if the input graph is not directed, |
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66 not connected or is a multigraph. |
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67 |
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68 NetworkXUnfeasible |
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69 This exception is raised in the following situations: |
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70 |
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71 * The sum of the demands is not zero. Then, there is no |
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72 flow satisfying all demands. |
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73 * There is no flow satisfying all demand. |
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74 |
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75 NetworkXUnbounded |
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76 This exception is raised if the digraph G has a cycle of |
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77 negative cost and infinite capacity. Then, the cost of a flow |
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78 satisfying all demands is unbounded below. |
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79 |
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80 Notes |
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81 ----- |
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82 This algorithm is not guaranteed to work if edge weights or demands |
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83 are floating point numbers (overflows and roundoff errors can |
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84 cause problems). As a workaround you can use integer numbers by |
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85 multiplying the relevant edge attributes by a convenient |
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86 constant factor (eg 100). |
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87 |
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88 See also |
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89 -------- |
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90 cost_of_flow, max_flow_min_cost, min_cost_flow, min_cost_flow_cost |
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91 |
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92 Examples |
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93 -------- |
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94 A simple example of a min cost flow problem. |
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95 |
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96 >>> G = nx.DiGraph() |
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97 >>> G.add_node("a", demand=-5) |
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98 >>> G.add_node("d", demand=5) |
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99 >>> G.add_edge("a", "b", weight=3, capacity=4) |
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100 >>> G.add_edge("a", "c", weight=6, capacity=10) |
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101 >>> G.add_edge("b", "d", weight=1, capacity=9) |
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102 >>> G.add_edge("c", "d", weight=2, capacity=5) |
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103 >>> flowCost, flowDict = nx.network_simplex(G) |
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104 >>> flowCost |
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105 24 |
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106 >>> flowDict # doctest: +SKIP |
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107 {'a': {'c': 1, 'b': 4}, 'c': {'d': 1}, 'b': {'d': 4}, 'd': {}} |
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108 |
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109 The mincost flow algorithm can also be used to solve shortest path |
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110 problems. To find the shortest path between two nodes u and v, |
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111 give all edges an infinite capacity, give node u a demand of -1 and |
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112 node v a demand a 1. Then run the network simplex. The value of a |
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113 min cost flow will be the distance between u and v and edges |
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114 carrying positive flow will indicate the path. |
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115 |
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116 >>> G = nx.DiGraph() |
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117 >>> G.add_weighted_edges_from( |
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118 ... [ |
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119 ... ("s", "u", 10), |
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120 ... ("s", "x", 5), |
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121 ... ("u", "v", 1), |
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122 ... ("u", "x", 2), |
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123 ... ("v", "y", 1), |
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124 ... ("x", "u", 3), |
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125 ... ("x", "v", 5), |
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126 ... ("x", "y", 2), |
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127 ... ("y", "s", 7), |
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128 ... ("y", "v", 6), |
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129 ... ] |
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130 ... ) |
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131 >>> G.add_node("s", demand=-1) |
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132 >>> G.add_node("v", demand=1) |
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133 >>> flowCost, flowDict = nx.network_simplex(G) |
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134 >>> flowCost == nx.shortest_path_length(G, "s", "v", weight="weight") |
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135 True |
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136 >>> sorted([(u, v) for u in flowDict for v in flowDict[u] if flowDict[u][v] > 0]) |
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137 [('s', 'x'), ('u', 'v'), ('x', 'u')] |
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138 >>> nx.shortest_path(G, "s", "v", weight="weight") |
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139 ['s', 'x', 'u', 'v'] |
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140 |
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141 It is possible to change the name of the attributes used for the |
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142 algorithm. |
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143 |
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144 >>> G = nx.DiGraph() |
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145 >>> G.add_node("p", spam=-4) |
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146 >>> G.add_node("q", spam=2) |
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147 >>> G.add_node("a", spam=-2) |
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148 >>> G.add_node("d", spam=-1) |
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149 >>> G.add_node("t", spam=2) |
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150 >>> G.add_node("w", spam=3) |
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151 >>> G.add_edge("p", "q", cost=7, vacancies=5) |
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152 >>> G.add_edge("p", "a", cost=1, vacancies=4) |
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153 >>> G.add_edge("q", "d", cost=2, vacancies=3) |
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154 >>> G.add_edge("t", "q", cost=1, vacancies=2) |
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155 >>> G.add_edge("a", "t", cost=2, vacancies=4) |
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156 >>> G.add_edge("d", "w", cost=3, vacancies=4) |
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157 >>> G.add_edge("t", "w", cost=4, vacancies=1) |
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158 >>> flowCost, flowDict = nx.network_simplex( |
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159 ... G, demand="spam", capacity="vacancies", weight="cost" |
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160 ... ) |
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161 >>> flowCost |
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162 37 |
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163 >>> flowDict # doctest: +SKIP |
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164 {'a': {'t': 4}, 'd': {'w': 2}, 'q': {'d': 1}, 'p': {'q': 2, 'a': 2}, 't': {'q': 1, 'w': 1}, 'w': {}} |
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165 |
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166 References |
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167 ---------- |
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168 .. [1] Z. Kiraly, P. Kovacs. |
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169 Efficient implementation of minimum-cost flow algorithms. |
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170 Acta Universitatis Sapientiae, Informatica 4(1):67--118. 2012. |
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171 .. [2] R. Barr, F. Glover, D. Klingman. |
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172 Enhancement of spanning tree labeling procedures for network |
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173 optimization. |
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174 INFOR 17(1):16--34. 1979. |
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175 """ |
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176 ########################################################################### |
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177 # Problem essentials extraction and sanity check |
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178 ########################################################################### |
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179 |
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180 if len(G) == 0: |
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181 raise nx.NetworkXError("graph has no nodes") |
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182 |
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183 # Number all nodes and edges and hereafter reference them using ONLY their |
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184 # numbers |
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185 |
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186 N = list(G) # nodes |
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187 I = {u: i for i, u in enumerate(N)} # node indices |
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188 D = [G.nodes[u].get(demand, 0) for u in N] # node demands |
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189 |
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190 inf = float("inf") |
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191 for p, b in zip(N, D): |
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192 if abs(b) == inf: |
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193 raise nx.NetworkXError(f"node {p!r} has infinite demand") |
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194 |
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195 multigraph = G.is_multigraph() |
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196 S = [] # edge sources |
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197 T = [] # edge targets |
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198 if multigraph: |
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199 K = [] # edge keys |
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200 E = {} # edge indices |
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201 U = [] # edge capacities |
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202 C = [] # edge weights |
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203 |
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204 if not multigraph: |
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205 edges = G.edges(data=True) |
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206 else: |
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207 edges = G.edges(data=True, keys=True) |
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208 edges = (e for e in edges if e[0] != e[1] and e[-1].get(capacity, inf) != 0) |
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209 for i, e in enumerate(edges): |
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210 S.append(I[e[0]]) |
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211 T.append(I[e[1]]) |
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212 if multigraph: |
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213 K.append(e[2]) |
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214 E[e[:-1]] = i |
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215 U.append(e[-1].get(capacity, inf)) |
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216 C.append(e[-1].get(weight, 0)) |
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217 |
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218 for e, c in zip(E, C): |
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219 if abs(c) == inf: |
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220 raise nx.NetworkXError(f"edge {e!r} has infinite weight") |
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221 if not multigraph: |
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222 edges = nx.selfloop_edges(G, data=True) |
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223 else: |
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224 edges = nx.selfloop_edges(G, data=True, keys=True) |
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225 for e in edges: |
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226 if abs(e[-1].get(weight, 0)) == inf: |
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227 raise nx.NetworkXError(f"edge {e[:-1]!r} has infinite weight") |
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228 |
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229 ########################################################################### |
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230 # Quick infeasibility detection |
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231 ########################################################################### |
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232 |
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233 if sum(D) != 0: |
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234 raise nx.NetworkXUnfeasible("total node demand is not zero") |
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235 for e, u in zip(E, U): |
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236 if u < 0: |
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237 raise nx.NetworkXUnfeasible(f"edge {e!r} has negative capacity") |
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238 if not multigraph: |
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239 edges = nx.selfloop_edges(G, data=True) |
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240 else: |
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241 edges = nx.selfloop_edges(G, data=True, keys=True) |
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242 for e in edges: |
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243 if e[-1].get(capacity, inf) < 0: |
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244 raise nx.NetworkXUnfeasible(f"edge {e[:-1]!r} has negative capacity") |
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245 |
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246 ########################################################################### |
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247 # Initialization |
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248 ########################################################################### |
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249 |
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250 # Add a dummy node -1 and connect all existing nodes to it with infinite- |
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251 # capacity dummy edges. Node -1 will serve as the root of the |
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252 # spanning tree of the network simplex method. The new edges will used to |
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253 # trivially satisfy the node demands and create an initial strongly |
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254 # feasible spanning tree. |
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255 n = len(N) # number of nodes |
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256 for p, d in enumerate(D): |
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257 # Must be greater-than here. Zero-demand nodes must have |
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258 # edges pointing towards the root to ensure strong |
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259 # feasibility. |
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260 if d > 0: |
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261 S.append(-1) |
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262 T.append(p) |
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263 else: |
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264 S.append(p) |
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265 T.append(-1) |
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266 faux_inf = ( |
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267 3 |
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268 * max( |
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269 chain( |
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270 [sum(u for u in U if u < inf), sum(abs(c) for c in C)], |
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271 (abs(d) for d in D), |
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272 ) |
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273 ) |
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274 or 1 |
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275 ) |
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276 C.extend(repeat(faux_inf, n)) |
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277 U.extend(repeat(faux_inf, n)) |
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278 |
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279 # Construct the initial spanning tree. |
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280 e = len(E) # number of edges |
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281 x = list(chain(repeat(0, e), (abs(d) for d in D))) # edge flows |
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282 pi = [faux_inf if d <= 0 else -faux_inf for d in D] # node potentials |
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283 parent = list(chain(repeat(-1, n), [None])) # parent nodes |
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284 edge = list(range(e, e + n)) # edges to parents |
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285 size = list(chain(repeat(1, n), [n + 1])) # subtree sizes |
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286 next = list(chain(range(1, n), [-1, 0])) # next nodes in depth-first thread |
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287 prev = list(range(-1, n)) # previous nodes in depth-first thread |
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288 last = list(chain(range(n), [n - 1])) # last descendants in depth-first thread |
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289 |
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290 ########################################################################### |
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291 # Pivot loop |
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292 ########################################################################### |
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293 |
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294 def reduced_cost(i): |
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295 """Returns the reduced cost of an edge i. |
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296 """ |
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297 c = C[i] - pi[S[i]] + pi[T[i]] |
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298 return c if x[i] == 0 else -c |
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299 |
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300 def find_entering_edges(): |
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301 """Yield entering edges until none can be found. |
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302 """ |
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303 if e == 0: |
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304 return |
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305 |
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306 # Entering edges are found by combining Dantzig's rule and Bland's |
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307 # rule. The edges are cyclically grouped into blocks of size B. Within |
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308 # each block, Dantzig's rule is applied to find an entering edge. The |
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309 # blocks to search is determined following Bland's rule. |
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310 B = int(ceil(sqrt(e))) # pivot block size |
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311 M = (e + B - 1) // B # number of blocks needed to cover all edges |
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312 m = 0 # number of consecutive blocks without eligible |
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313 # entering edges |
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314 f = 0 # first edge in block |
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315 while m < M: |
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316 # Determine the next block of edges. |
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317 l = f + B |
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318 if l <= e: |
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319 edges = range(f, l) |
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320 else: |
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321 l -= e |
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322 edges = chain(range(f, e), range(l)) |
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323 f = l |
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324 # Find the first edge with the lowest reduced cost. |
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325 i = min(edges, key=reduced_cost) |
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326 c = reduced_cost(i) |
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327 if c >= 0: |
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328 # No entering edge found in the current block. |
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329 m += 1 |
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330 else: |
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331 # Entering edge found. |
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332 if x[i] == 0: |
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333 p = S[i] |
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334 q = T[i] |
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335 else: |
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336 p = T[i] |
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337 q = S[i] |
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338 yield i, p, q |
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339 m = 0 |
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340 # All edges have nonnegative reduced costs. The current flow is |
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341 # optimal. |
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342 |
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343 def find_apex(p, q): |
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344 """Find the lowest common ancestor of nodes p and q in the spanning |
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345 tree. |
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346 """ |
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347 size_p = size[p] |
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348 size_q = size[q] |
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349 while True: |
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350 while size_p < size_q: |
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351 p = parent[p] |
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352 size_p = size[p] |
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353 while size_p > size_q: |
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354 q = parent[q] |
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355 size_q = size[q] |
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356 if size_p == size_q: |
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357 if p != q: |
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358 p = parent[p] |
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359 size_p = size[p] |
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360 q = parent[q] |
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361 size_q = size[q] |
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362 else: |
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363 return p |
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364 |
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365 def trace_path(p, w): |
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366 """Returns the nodes and edges on the path from node p to its ancestor |
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367 w. |
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368 """ |
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369 Wn = [p] |
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370 We = [] |
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371 while p != w: |
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372 We.append(edge[p]) |
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373 p = parent[p] |
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374 Wn.append(p) |
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375 return Wn, We |
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376 |
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377 def find_cycle(i, p, q): |
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378 """Returns the nodes and edges on the cycle containing edge i == (p, q) |
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379 when the latter is added to the spanning tree. |
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380 |
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381 The cycle is oriented in the direction from p to q. |
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382 """ |
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383 w = find_apex(p, q) |
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384 Wn, We = trace_path(p, w) |
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385 Wn.reverse() |
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386 We.reverse() |
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387 if We != [i]: |
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388 We.append(i) |
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389 WnR, WeR = trace_path(q, w) |
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390 del WnR[-1] |
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391 Wn += WnR |
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392 We += WeR |
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393 return Wn, We |
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394 |
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395 def residual_capacity(i, p): |
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396 """Returns the residual capacity of an edge i in the direction away |
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397 from its endpoint p. |
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398 """ |
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399 return U[i] - x[i] if S[i] == p else x[i] |
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400 |
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401 def find_leaving_edge(Wn, We): |
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402 """Returns the leaving edge in a cycle represented by Wn and We. |
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403 """ |
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404 j, s = min( |
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405 zip(reversed(We), reversed(Wn)), key=lambda i_p: residual_capacity(*i_p) |
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406 ) |
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407 t = T[j] if S[j] == s else S[j] |
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408 return j, s, t |
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409 |
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410 def augment_flow(Wn, We, f): |
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411 """Augment f units of flow along a cycle represented by Wn and We. |
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412 """ |
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413 for i, p in zip(We, Wn): |
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414 if S[i] == p: |
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415 x[i] += f |
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416 else: |
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417 x[i] -= f |
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418 |
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419 def trace_subtree(p): |
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420 """Yield the nodes in the subtree rooted at a node p. |
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421 """ |
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422 yield p |
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423 l = last[p] |
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424 while p != l: |
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425 p = next[p] |
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426 yield p |
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427 |
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428 def remove_edge(s, t): |
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429 """Remove an edge (s, t) where parent[t] == s from the spanning tree. |
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430 """ |
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431 size_t = size[t] |
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432 prev_t = prev[t] |
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433 last_t = last[t] |
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434 next_last_t = next[last_t] |
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435 # Remove (s, t). |
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436 parent[t] = None |
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437 edge[t] = None |
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438 # Remove the subtree rooted at t from the depth-first thread. |
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439 next[prev_t] = next_last_t |
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440 prev[next_last_t] = prev_t |
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441 next[last_t] = t |
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442 prev[t] = last_t |
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443 # Update the subtree sizes and last descendants of the (old) acenstors |
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444 # of t. |
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445 while s is not None: |
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446 size[s] -= size_t |
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447 if last[s] == last_t: |
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448 last[s] = prev_t |
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449 s = parent[s] |
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450 |
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451 def make_root(q): |
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452 """Make a node q the root of its containing subtree. |
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453 """ |
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454 ancestors = [] |
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455 while q is not None: |
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456 ancestors.append(q) |
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457 q = parent[q] |
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458 ancestors.reverse() |
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459 for p, q in zip(ancestors, islice(ancestors, 1, None)): |
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460 size_p = size[p] |
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461 last_p = last[p] |
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462 prev_q = prev[q] |
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463 last_q = last[q] |
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464 next_last_q = next[last_q] |
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465 # Make p a child of q. |
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466 parent[p] = q |
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467 parent[q] = None |
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468 edge[p] = edge[q] |
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469 edge[q] = None |
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470 size[p] = size_p - size[q] |
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471 size[q] = size_p |
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472 # Remove the subtree rooted at q from the depth-first thread. |
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473 next[prev_q] = next_last_q |
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474 prev[next_last_q] = prev_q |
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475 next[last_q] = q |
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476 prev[q] = last_q |
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477 if last_p == last_q: |
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478 last[p] = prev_q |
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479 last_p = prev_q |
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480 # Add the remaining parts of the subtree rooted at p as a subtree |
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481 # of q in the depth-first thread. |
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482 prev[p] = last_q |
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483 next[last_q] = p |
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484 next[last_p] = q |
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485 prev[q] = last_p |
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486 last[q] = last_p |
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487 |
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488 def add_edge(i, p, q): |
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489 """Add an edge (p, q) to the spanning tree where q is the root of a |
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490 subtree. |
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491 """ |
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492 last_p = last[p] |
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493 next_last_p = next[last_p] |
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494 size_q = size[q] |
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495 last_q = last[q] |
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496 # Make q a child of p. |
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497 parent[q] = p |
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498 edge[q] = i |
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499 # Insert the subtree rooted at q into the depth-first thread. |
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500 next[last_p] = q |
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501 prev[q] = last_p |
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502 prev[next_last_p] = last_q |
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503 next[last_q] = next_last_p |
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504 # Update the subtree sizes and last descendants of the (new) ancestors |
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505 # of q. |
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506 while p is not None: |
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507 size[p] += size_q |
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508 if last[p] == last_p: |
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509 last[p] = last_q |
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510 p = parent[p] |
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511 |
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512 def update_potentials(i, p, q): |
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513 """Update the potentials of the nodes in the subtree rooted at a node |
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514 q connected to its parent p by an edge i. |
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515 """ |
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516 if q == T[i]: |
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517 d = pi[p] - C[i] - pi[q] |
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518 else: |
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519 d = pi[p] + C[i] - pi[q] |
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520 for q in trace_subtree(q): |
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521 pi[q] += d |
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522 |
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523 # Pivot loop |
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524 for i, p, q in find_entering_edges(): |
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525 Wn, We = find_cycle(i, p, q) |
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526 j, s, t = find_leaving_edge(Wn, We) |
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527 augment_flow(Wn, We, residual_capacity(j, s)) |
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528 # Do nothing more if the entering edge is the same as the leaving edge. |
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529 if i != j: |
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530 if parent[t] != s: |
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531 # Ensure that s is the parent of t. |
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532 s, t = t, s |
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533 if We.index(i) > We.index(j): |
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534 # Ensure that q is in the subtree rooted at t. |
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535 p, q = q, p |
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536 remove_edge(s, t) |
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537 make_root(q) |
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538 add_edge(i, p, q) |
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539 update_potentials(i, p, q) |
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540 |
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541 ########################################################################### |
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542 # Infeasibility and unboundedness detection |
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543 ########################################################################### |
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544 |
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545 if any(x[i] != 0 for i in range(-n, 0)): |
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546 raise nx.NetworkXUnfeasible("no flow satisfies all node demands") |
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547 |
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548 if any(x[i] * 2 >= faux_inf for i in range(e)) or any( |
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549 e[-1].get(capacity, inf) == inf and e[-1].get(weight, 0) < 0 |
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550 for e in nx.selfloop_edges(G, data=True) |
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551 ): |
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552 raise nx.NetworkXUnbounded("negative cycle with infinite capacity found") |
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553 |
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554 ########################################################################### |
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555 # Flow cost calculation and flow dict construction |
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556 ########################################################################### |
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557 |
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558 del x[e:] |
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559 flow_cost = sum(c * x for c, x in zip(C, x)) |
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560 flow_dict = {n: {} for n in N} |
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561 |
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562 def add_entry(e): |
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563 """Add a flow dict entry. |
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564 """ |
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565 d = flow_dict[e[0]] |
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566 for k in e[1:-2]: |
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567 try: |
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568 d = d[k] |
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569 except KeyError: |
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570 t = {} |
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571 d[k] = t |
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572 d = t |
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573 d[e[-2]] = e[-1] |
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574 |
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575 S = (N[s] for s in S) # Use original nodes. |
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576 T = (N[t] for t in T) # Use original nodes. |
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577 if not multigraph: |
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578 for e in zip(S, T, x): |
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579 add_entry(e) |
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580 edges = G.edges(data=True) |
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581 else: |
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582 for e in zip(S, T, K, x): |
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583 add_entry(e) |
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584 edges = G.edges(data=True, keys=True) |
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585 for e in edges: |
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586 if e[0] != e[1]: |
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587 if e[-1].get(capacity, inf) == 0: |
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588 add_entry(e[:-1] + (0,)) |
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589 else: |
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590 c = e[-1].get(weight, 0) |
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591 if c >= 0: |
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592 add_entry(e[:-1] + (0,)) |
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593 else: |
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594 u = e[-1][capacity] |
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595 flow_cost += c * u |
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596 add_entry(e[:-1] + (u,)) |
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597 |
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598 return flow_cost, flow_dict |