Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/flow/shortestaugmentingpath.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 Shortest augmenting path algorithm for maximum flow problems. |
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3 """ |
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4 |
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5 from collections import deque |
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6 import networkx as nx |
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7 from .utils import build_residual_network, CurrentEdge |
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8 from .edmondskarp import edmonds_karp_core |
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9 |
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10 __all__ = ["shortest_augmenting_path"] |
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11 |
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12 |
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13 def shortest_augmenting_path_impl(G, s, t, capacity, residual, two_phase, cutoff): |
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14 """Implementation of the shortest augmenting path algorithm. |
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15 """ |
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16 if s not in G: |
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17 raise nx.NetworkXError(f"node {str(s)} not in graph") |
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18 if t not in G: |
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19 raise nx.NetworkXError(f"node {str(t)} not in graph") |
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20 if s == t: |
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21 raise nx.NetworkXError("source and sink are the same node") |
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22 |
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23 if residual is None: |
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24 R = build_residual_network(G, capacity) |
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25 else: |
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26 R = residual |
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27 |
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28 R_nodes = R.nodes |
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29 R_pred = R.pred |
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30 R_succ = R.succ |
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31 |
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32 # Initialize/reset the residual network. |
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33 for u in R: |
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34 for e in R_succ[u].values(): |
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35 e["flow"] = 0 |
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36 |
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37 # Initialize heights of the nodes. |
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38 heights = {t: 0} |
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39 q = deque([(t, 0)]) |
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40 while q: |
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41 u, height = q.popleft() |
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42 height += 1 |
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43 for v, attr in R_pred[u].items(): |
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44 if v not in heights and attr["flow"] < attr["capacity"]: |
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45 heights[v] = height |
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46 q.append((v, height)) |
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47 |
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48 if s not in heights: |
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49 # t is not reachable from s in the residual network. The maximum flow |
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50 # must be zero. |
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51 R.graph["flow_value"] = 0 |
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52 return R |
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53 |
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54 n = len(G) |
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55 m = R.size() / 2 |
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56 |
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57 # Initialize heights and 'current edge' data structures of the nodes. |
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58 for u in R: |
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59 R_nodes[u]["height"] = heights[u] if u in heights else n |
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60 R_nodes[u]["curr_edge"] = CurrentEdge(R_succ[u]) |
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61 |
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62 # Initialize counts of nodes in each level. |
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63 counts = [0] * (2 * n - 1) |
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64 for u in R: |
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65 counts[R_nodes[u]["height"]] += 1 |
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66 |
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67 inf = R.graph["inf"] |
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68 |
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69 def augment(path): |
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70 """Augment flow along a path from s to t. |
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71 """ |
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72 # Determine the path residual capacity. |
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73 flow = inf |
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74 it = iter(path) |
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75 u = next(it) |
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76 for v in it: |
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77 attr = R_succ[u][v] |
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78 flow = min(flow, attr["capacity"] - attr["flow"]) |
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79 u = v |
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80 if flow * 2 > inf: |
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81 raise nx.NetworkXUnbounded("Infinite capacity path, flow unbounded above.") |
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82 # Augment flow along the path. |
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83 it = iter(path) |
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84 u = next(it) |
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85 for v in it: |
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86 R_succ[u][v]["flow"] += flow |
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87 R_succ[v][u]["flow"] -= flow |
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88 u = v |
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89 return flow |
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90 |
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91 def relabel(u): |
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92 """Relabel a node to create an admissible edge. |
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93 """ |
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94 height = n - 1 |
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95 for v, attr in R_succ[u].items(): |
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96 if attr["flow"] < attr["capacity"]: |
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97 height = min(height, R_nodes[v]["height"]) |
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98 return height + 1 |
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99 |
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100 if cutoff is None: |
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101 cutoff = float("inf") |
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102 |
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103 # Phase 1: Look for shortest augmenting paths using depth-first search. |
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104 |
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105 flow_value = 0 |
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106 path = [s] |
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107 u = s |
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108 d = n if not two_phase else int(min(m ** 0.5, 2 * n ** (2.0 / 3))) |
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109 done = R_nodes[s]["height"] >= d |
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110 while not done: |
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111 height = R_nodes[u]["height"] |
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112 curr_edge = R_nodes[u]["curr_edge"] |
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113 # Depth-first search for the next node on the path to t. |
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114 while True: |
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115 v, attr = curr_edge.get() |
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116 if height == R_nodes[v]["height"] + 1 and attr["flow"] < attr["capacity"]: |
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117 # Advance to the next node following an admissible edge. |
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118 path.append(v) |
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119 u = v |
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120 break |
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121 try: |
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122 curr_edge.move_to_next() |
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123 except StopIteration: |
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124 counts[height] -= 1 |
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125 if counts[height] == 0: |
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126 # Gap heuristic: If relabeling causes a level to become |
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127 # empty, a minimum cut has been identified. The algorithm |
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128 # can now be terminated. |
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129 R.graph["flow_value"] = flow_value |
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130 return R |
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131 height = relabel(u) |
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132 if u == s and height >= d: |
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133 if not two_phase: |
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134 # t is disconnected from s in the residual network. No |
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135 # more augmenting paths exist. |
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136 R.graph["flow_value"] = flow_value |
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137 return R |
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138 else: |
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139 # t is at least d steps away from s. End of phase 1. |
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140 done = True |
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141 break |
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142 counts[height] += 1 |
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143 R_nodes[u]["height"] = height |
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144 if u != s: |
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145 # After relabeling, the last edge on the path is no longer |
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146 # admissible. Retreat one step to look for an alternative. |
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147 path.pop() |
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148 u = path[-1] |
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149 break |
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150 if u == t: |
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151 # t is reached. Augment flow along the path and reset it for a new |
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152 # depth-first search. |
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153 flow_value += augment(path) |
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154 if flow_value >= cutoff: |
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155 R.graph["flow_value"] = flow_value |
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156 return R |
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157 path = [s] |
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158 u = s |
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159 |
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160 # Phase 2: Look for shortest augmenting paths using breadth-first search. |
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161 flow_value += edmonds_karp_core(R, s, t, cutoff - flow_value) |
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162 |
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163 R.graph["flow_value"] = flow_value |
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164 return R |
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165 |
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166 |
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167 def shortest_augmenting_path( |
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168 G, |
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169 s, |
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170 t, |
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171 capacity="capacity", |
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172 residual=None, |
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173 value_only=False, |
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174 two_phase=False, |
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175 cutoff=None, |
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176 ): |
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177 r"""Find a maximum single-commodity flow using the shortest augmenting path |
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178 algorithm. |
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179 |
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180 This function returns the residual network resulting after computing |
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181 the maximum flow. See below for details about the conventions |
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182 NetworkX uses for defining residual networks. |
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183 |
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184 This algorithm has a running time of $O(n^2 m)$ for $n$ nodes and $m$ |
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185 edges. |
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186 |
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187 |
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188 Parameters |
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189 ---------- |
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190 G : NetworkX graph |
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191 Edges of the graph are expected to have an attribute called |
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192 'capacity'. If this attribute is not present, the edge is |
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193 considered to have infinite capacity. |
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194 |
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195 s : node |
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196 Source node for the flow. |
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197 |
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198 t : node |
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199 Sink node for the flow. |
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200 |
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201 capacity : string |
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202 Edges of the graph G are expected to have an attribute capacity |
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203 that indicates how much flow the edge can support. If this |
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204 attribute is not present, the edge is considered to have |
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205 infinite capacity. Default value: 'capacity'. |
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206 |
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207 residual : NetworkX graph |
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208 Residual network on which the algorithm is to be executed. If None, a |
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209 new residual network is created. Default value: None. |
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210 |
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211 value_only : bool |
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212 If True compute only the value of the maximum flow. This parameter |
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213 will be ignored by this algorithm because it is not applicable. |
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214 |
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215 two_phase : bool |
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216 If True, a two-phase variant is used. The two-phase variant improves |
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217 the running time on unit-capacity networks from $O(nm)$ to |
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218 $O(\min(n^{2/3}, m^{1/2}) m)$. Default value: False. |
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219 |
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220 cutoff : integer, float |
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221 If specified, the algorithm will terminate when the flow value reaches |
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222 or exceeds the cutoff. In this case, it may be unable to immediately |
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223 determine a minimum cut. Default value: None. |
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224 |
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225 Returns |
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226 ------- |
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227 R : NetworkX DiGraph |
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228 Residual network after computing the maximum flow. |
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229 |
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230 Raises |
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231 ------ |
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232 NetworkXError |
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233 The algorithm does not support MultiGraph and MultiDiGraph. If |
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234 the input graph is an instance of one of these two classes, a |
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235 NetworkXError is raised. |
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236 |
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237 NetworkXUnbounded |
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238 If the graph has a path of infinite capacity, the value of a |
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239 feasible flow on the graph is unbounded above and the function |
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240 raises a NetworkXUnbounded. |
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241 |
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242 See also |
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243 -------- |
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244 :meth:`maximum_flow` |
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245 :meth:`minimum_cut` |
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246 :meth:`edmonds_karp` |
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247 :meth:`preflow_push` |
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248 |
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249 Notes |
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250 ----- |
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251 The residual network :samp:`R` from an input graph :samp:`G` has the |
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252 same nodes as :samp:`G`. :samp:`R` is a DiGraph that contains a pair |
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253 of edges :samp:`(u, v)` and :samp:`(v, u)` iff :samp:`(u, v)` is not a |
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254 self-loop, and at least one of :samp:`(u, v)` and :samp:`(v, u)` exists |
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255 in :samp:`G`. |
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256 |
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257 For each edge :samp:`(u, v)` in :samp:`R`, :samp:`R[u][v]['capacity']` |
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258 is equal to the capacity of :samp:`(u, v)` in :samp:`G` if it exists |
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259 in :samp:`G` or zero otherwise. If the capacity is infinite, |
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260 :samp:`R[u][v]['capacity']` will have a high arbitrary finite value |
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261 that does not affect the solution of the problem. This value is stored in |
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262 :samp:`R.graph['inf']`. For each edge :samp:`(u, v)` in :samp:`R`, |
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263 :samp:`R[u][v]['flow']` represents the flow function of :samp:`(u, v)` and |
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264 satisfies :samp:`R[u][v]['flow'] == -R[v][u]['flow']`. |
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265 |
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266 The flow value, defined as the total flow into :samp:`t`, the sink, is |
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267 stored in :samp:`R.graph['flow_value']`. If :samp:`cutoff` is not |
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268 specified, reachability to :samp:`t` using only edges :samp:`(u, v)` such |
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269 that :samp:`R[u][v]['flow'] < R[u][v]['capacity']` induces a minimum |
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270 :samp:`s`-:samp:`t` cut. |
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271 |
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272 Examples |
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273 -------- |
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274 >>> from networkx.algorithms.flow import shortest_augmenting_path |
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275 |
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276 The functions that implement flow algorithms and output a residual |
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277 network, such as this one, are not imported to the base NetworkX |
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278 namespace, so you have to explicitly import them from the flow package. |
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279 |
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280 >>> G = nx.DiGraph() |
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281 >>> G.add_edge("x", "a", capacity=3.0) |
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282 >>> G.add_edge("x", "b", capacity=1.0) |
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283 >>> G.add_edge("a", "c", capacity=3.0) |
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284 >>> G.add_edge("b", "c", capacity=5.0) |
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285 >>> G.add_edge("b", "d", capacity=4.0) |
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286 >>> G.add_edge("d", "e", capacity=2.0) |
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287 >>> G.add_edge("c", "y", capacity=2.0) |
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288 >>> G.add_edge("e", "y", capacity=3.0) |
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289 >>> R = shortest_augmenting_path(G, "x", "y") |
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290 >>> flow_value = nx.maximum_flow_value(G, "x", "y") |
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291 >>> flow_value |
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292 3.0 |
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293 >>> flow_value == R.graph["flow_value"] |
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294 True |
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295 |
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296 """ |
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297 R = shortest_augmenting_path_impl(G, s, t, capacity, residual, two_phase, cutoff) |
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298 R.graph["algorithm"] = "shortest_augmenting_path" |
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299 return R |