Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/flow/preflowpush.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 Highest-label preflow-push 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 from itertools import islice |
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7 import networkx as nx |
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8 from ...utils import arbitrary_element |
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9 from .utils import build_residual_network |
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10 from .utils import CurrentEdge |
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11 from .utils import detect_unboundedness |
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12 from .utils import GlobalRelabelThreshold |
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13 from .utils import Level |
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14 |
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15 __all__ = ["preflow_push"] |
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16 |
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17 |
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18 def preflow_push_impl(G, s, t, capacity, residual, global_relabel_freq, value_only): |
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19 """Implementation of the highest-label preflow-push algorithm. |
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20 """ |
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21 if s not in G: |
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22 raise nx.NetworkXError(f"node {str(s)} not in graph") |
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23 if t not in G: |
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24 raise nx.NetworkXError(f"node {str(t)} not in graph") |
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25 if s == t: |
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26 raise nx.NetworkXError("source and sink are the same node") |
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27 |
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28 if global_relabel_freq is None: |
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29 global_relabel_freq = 0 |
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30 if global_relabel_freq < 0: |
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31 raise nx.NetworkXError("global_relabel_freq must be nonnegative.") |
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32 |
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33 if residual is None: |
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34 R = build_residual_network(G, capacity) |
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35 else: |
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36 R = residual |
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37 |
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38 detect_unboundedness(R, s, t) |
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39 |
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40 R_nodes = R.nodes |
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41 R_pred = R.pred |
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42 R_succ = R.succ |
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43 |
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44 # Initialize/reset the residual network. |
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45 for u in R: |
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46 R_nodes[u]["excess"] = 0 |
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47 for e in R_succ[u].values(): |
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48 e["flow"] = 0 |
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49 |
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50 def reverse_bfs(src): |
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51 """Perform a reverse breadth-first search from src in the residual |
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52 network. |
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53 """ |
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54 heights = {src: 0} |
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55 q = deque([(src, 0)]) |
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56 while q: |
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57 u, height = q.popleft() |
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58 height += 1 |
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59 for v, attr in R_pred[u].items(): |
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60 if v not in heights and attr["flow"] < attr["capacity"]: |
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61 heights[v] = height |
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62 q.append((v, height)) |
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63 return heights |
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64 |
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65 # Initialize heights of the nodes. |
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66 heights = reverse_bfs(t) |
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67 |
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68 if s not in heights: |
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69 # t is not reachable from s in the residual network. The maximum flow |
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70 # must be zero. |
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71 R.graph["flow_value"] = 0 |
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72 return R |
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73 |
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74 n = len(R) |
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75 # max_height represents the height of the highest level below level n with |
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76 # at least one active node. |
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77 max_height = max(heights[u] for u in heights if u != s) |
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78 heights[s] = n |
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79 |
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80 grt = GlobalRelabelThreshold(n, R.size(), global_relabel_freq) |
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81 |
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82 # Initialize heights and 'current edge' data structures of the nodes. |
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83 for u in R: |
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84 R_nodes[u]["height"] = heights[u] if u in heights else n + 1 |
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85 R_nodes[u]["curr_edge"] = CurrentEdge(R_succ[u]) |
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86 |
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87 def push(u, v, flow): |
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88 """Push flow units of flow from u to v. |
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89 """ |
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90 R_succ[u][v]["flow"] += flow |
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91 R_succ[v][u]["flow"] -= flow |
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92 R_nodes[u]["excess"] -= flow |
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93 R_nodes[v]["excess"] += flow |
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94 |
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95 # The maximum flow must be nonzero now. Initialize the preflow by |
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96 # saturating all edges emanating from s. |
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97 for u, attr in R_succ[s].items(): |
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98 flow = attr["capacity"] |
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99 if flow > 0: |
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100 push(s, u, flow) |
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101 |
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102 # Partition nodes into levels. |
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103 levels = [Level() for i in range(2 * n)] |
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104 for u in R: |
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105 if u != s and u != t: |
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106 level = levels[R_nodes[u]["height"]] |
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107 if R_nodes[u]["excess"] > 0: |
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108 level.active.add(u) |
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109 else: |
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110 level.inactive.add(u) |
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111 |
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112 def activate(v): |
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113 """Move a node from the inactive set to the active set of its level. |
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114 """ |
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115 if v != s and v != t: |
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116 level = levels[R_nodes[v]["height"]] |
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117 if v in level.inactive: |
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118 level.inactive.remove(v) |
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119 level.active.add(v) |
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120 |
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121 def relabel(u): |
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122 """Relabel a node to create an admissible edge. |
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123 """ |
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124 grt.add_work(len(R_succ[u])) |
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125 return ( |
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126 min( |
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127 R_nodes[v]["height"] |
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128 for v, attr in R_succ[u].items() |
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129 if attr["flow"] < attr["capacity"] |
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130 ) |
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131 + 1 |
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132 ) |
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133 |
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134 def discharge(u, is_phase1): |
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135 """Discharge a node until it becomes inactive or, during phase 1 (see |
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136 below), its height reaches at least n. The node is known to have the |
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137 largest height among active nodes. |
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138 """ |
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139 height = R_nodes[u]["height"] |
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140 curr_edge = R_nodes[u]["curr_edge"] |
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141 # next_height represents the next height to examine after discharging |
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142 # the current node. During phase 1, it is capped to below n. |
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143 next_height = height |
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144 levels[height].active.remove(u) |
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145 while True: |
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146 v, attr = curr_edge.get() |
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147 if height == R_nodes[v]["height"] + 1 and attr["flow"] < attr["capacity"]: |
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148 flow = min(R_nodes[u]["excess"], attr["capacity"] - attr["flow"]) |
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149 push(u, v, flow) |
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150 activate(v) |
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151 if R_nodes[u]["excess"] == 0: |
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152 # The node has become inactive. |
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153 levels[height].inactive.add(u) |
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154 break |
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155 try: |
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156 curr_edge.move_to_next() |
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157 except StopIteration: |
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158 # We have run off the end of the adjacency list, and there can |
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159 # be no more admissible edges. Relabel the node to create one. |
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160 height = relabel(u) |
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161 if is_phase1 and height >= n - 1: |
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162 # Although the node is still active, with a height at least |
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163 # n - 1, it is now known to be on the s side of the minimum |
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164 # s-t cut. Stop processing it until phase 2. |
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165 levels[height].active.add(u) |
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166 break |
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167 # The first relabel operation after global relabeling may not |
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168 # increase the height of the node since the 'current edge' data |
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169 # structure is not rewound. Use height instead of (height - 1) |
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170 # in case other active nodes at the same level are missed. |
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171 next_height = height |
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172 R_nodes[u]["height"] = height |
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173 return next_height |
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174 |
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175 def gap_heuristic(height): |
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176 """Apply the gap heuristic. |
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177 """ |
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178 # Move all nodes at levels (height + 1) to max_height to level n + 1. |
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179 for level in islice(levels, height + 1, max_height + 1): |
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180 for u in level.active: |
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181 R_nodes[u]["height"] = n + 1 |
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182 for u in level.inactive: |
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183 R_nodes[u]["height"] = n + 1 |
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184 levels[n + 1].active.update(level.active) |
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185 level.active.clear() |
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186 levels[n + 1].inactive.update(level.inactive) |
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187 level.inactive.clear() |
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188 |
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189 def global_relabel(from_sink): |
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190 """Apply the global relabeling heuristic. |
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191 """ |
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192 src = t if from_sink else s |
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193 heights = reverse_bfs(src) |
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194 if not from_sink: |
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195 # s must be reachable from t. Remove t explicitly. |
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196 del heights[t] |
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197 max_height = max(heights.values()) |
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198 if from_sink: |
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199 # Also mark nodes from which t is unreachable for relabeling. This |
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200 # serves the same purpose as the gap heuristic. |
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201 for u in R: |
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202 if u not in heights and R_nodes[u]["height"] < n: |
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203 heights[u] = n + 1 |
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204 else: |
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205 # Shift the computed heights because the height of s is n. |
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206 for u in heights: |
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207 heights[u] += n |
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208 max_height += n |
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209 del heights[src] |
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210 for u, new_height in heights.items(): |
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211 old_height = R_nodes[u]["height"] |
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212 if new_height != old_height: |
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213 if u in levels[old_height].active: |
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214 levels[old_height].active.remove(u) |
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215 levels[new_height].active.add(u) |
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216 else: |
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217 levels[old_height].inactive.remove(u) |
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218 levels[new_height].inactive.add(u) |
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219 R_nodes[u]["height"] = new_height |
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220 return max_height |
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221 |
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222 # Phase 1: Find the maximum preflow by pushing as much flow as possible to |
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223 # t. |
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224 |
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225 height = max_height |
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226 while height > 0: |
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227 # Discharge active nodes in the current level. |
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228 while True: |
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229 level = levels[height] |
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230 if not level.active: |
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231 # All active nodes in the current level have been discharged. |
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232 # Move to the next lower level. |
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233 height -= 1 |
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234 break |
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235 # Record the old height and level for the gap heuristic. |
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236 old_height = height |
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237 old_level = level |
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238 u = arbitrary_element(level.active) |
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239 height = discharge(u, True) |
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240 if grt.is_reached(): |
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241 # Global relabeling heuristic: Recompute the exact heights of |
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242 # all nodes. |
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243 height = global_relabel(True) |
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244 max_height = height |
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245 grt.clear_work() |
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246 elif not old_level.active and not old_level.inactive: |
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247 # Gap heuristic: If the level at old_height is empty (a 'gap'), |
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248 # a minimum cut has been identified. All nodes with heights |
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249 # above old_height can have their heights set to n + 1 and not |
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250 # be further processed before a maximum preflow is found. |
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251 gap_heuristic(old_height) |
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252 height = old_height - 1 |
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253 max_height = height |
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254 else: |
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255 # Update the height of the highest level with at least one |
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256 # active node. |
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257 max_height = max(max_height, height) |
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258 |
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259 # A maximum preflow has been found. The excess at t is the maximum flow |
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260 # value. |
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261 if value_only: |
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262 R.graph["flow_value"] = R_nodes[t]["excess"] |
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263 return R |
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264 |
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265 # Phase 2: Convert the maximum preflow into a maximum flow by returning the |
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266 # excess to s. |
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267 |
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268 # Relabel all nodes so that they have accurate heights. |
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269 height = global_relabel(False) |
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270 grt.clear_work() |
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271 |
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272 # Continue to discharge the active nodes. |
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273 while height > n: |
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274 # Discharge active nodes in the current level. |
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275 while True: |
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276 level = levels[height] |
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277 if not level.active: |
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278 # All active nodes in the current level have been discharged. |
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279 # Move to the next lower level. |
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280 height -= 1 |
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281 break |
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282 u = arbitrary_element(level.active) |
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283 height = discharge(u, False) |
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284 if grt.is_reached(): |
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285 # Global relabeling heuristic. |
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286 height = global_relabel(False) |
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287 grt.clear_work() |
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288 |
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289 R.graph["flow_value"] = R_nodes[t]["excess"] |
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290 return R |
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291 |
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292 |
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293 def preflow_push( |
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294 G, s, t, capacity="capacity", residual=None, global_relabel_freq=1, value_only=False |
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295 ): |
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296 r"""Find a maximum single-commodity flow using the highest-label |
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297 preflow-push algorithm. |
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298 |
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299 This function returns the residual network resulting after computing |
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300 the maximum flow. See below for details about the conventions |
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301 NetworkX uses for defining residual networks. |
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302 |
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303 This algorithm has a running time of $O(n^2 \sqrt{m})$ for $n$ nodes and |
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304 $m$ edges. |
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305 |
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306 |
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307 Parameters |
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308 ---------- |
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309 G : NetworkX graph |
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310 Edges of the graph are expected to have an attribute called |
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311 'capacity'. If this attribute is not present, the edge is |
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312 considered to have infinite capacity. |
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313 |
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314 s : node |
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315 Source node for the flow. |
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316 |
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317 t : node |
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318 Sink node for the flow. |
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319 |
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320 capacity : string |
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321 Edges of the graph G are expected to have an attribute capacity |
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322 that indicates how much flow the edge can support. If this |
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323 attribute is not present, the edge is considered to have |
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324 infinite capacity. Default value: 'capacity'. |
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325 |
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326 residual : NetworkX graph |
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327 Residual network on which the algorithm is to be executed. If None, a |
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328 new residual network is created. Default value: None. |
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329 |
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330 global_relabel_freq : integer, float |
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331 Relative frequency of applying the global relabeling heuristic to speed |
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332 up the algorithm. If it is None, the heuristic is disabled. Default |
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333 value: 1. |
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334 |
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335 value_only : bool |
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336 If False, compute a maximum flow; otherwise, compute a maximum preflow |
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337 which is enough for computing the maximum flow value. Default value: |
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338 False. |
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339 |
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340 Returns |
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341 ------- |
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342 R : NetworkX DiGraph |
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343 Residual network after computing the maximum flow. |
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344 |
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345 Raises |
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346 ------ |
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347 NetworkXError |
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348 The algorithm does not support MultiGraph and MultiDiGraph. If |
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349 the input graph is an instance of one of these two classes, a |
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350 NetworkXError is raised. |
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351 |
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352 NetworkXUnbounded |
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353 If the graph has a path of infinite capacity, the value of a |
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354 feasible flow on the graph is unbounded above and the function |
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355 raises a NetworkXUnbounded. |
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356 |
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357 See also |
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358 -------- |
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359 :meth:`maximum_flow` |
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360 :meth:`minimum_cut` |
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361 :meth:`edmonds_karp` |
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362 :meth:`shortest_augmenting_path` |
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363 |
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364 Notes |
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365 ----- |
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366 The residual network :samp:`R` from an input graph :samp:`G` has the |
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367 same nodes as :samp:`G`. :samp:`R` is a DiGraph that contains a pair |
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368 of edges :samp:`(u, v)` and :samp:`(v, u)` iff :samp:`(u, v)` is not a |
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369 self-loop, and at least one of :samp:`(u, v)` and :samp:`(v, u)` exists |
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370 in :samp:`G`. For each node :samp:`u` in :samp:`R`, |
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371 :samp:`R.nodes[u]['excess']` represents the difference between flow into |
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372 :samp:`u` and flow out of :samp:`u`. |
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373 |
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374 For each edge :samp:`(u, v)` in :samp:`R`, :samp:`R[u][v]['capacity']` |
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375 is equal to the capacity of :samp:`(u, v)` in :samp:`G` if it exists |
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376 in :samp:`G` or zero otherwise. If the capacity is infinite, |
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377 :samp:`R[u][v]['capacity']` will have a high arbitrary finite value |
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378 that does not affect the solution of the problem. This value is stored in |
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379 :samp:`R.graph['inf']`. For each edge :samp:`(u, v)` in :samp:`R`, |
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380 :samp:`R[u][v]['flow']` represents the flow function of :samp:`(u, v)` and |
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381 satisfies :samp:`R[u][v]['flow'] == -R[v][u]['flow']`. |
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382 |
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383 The flow value, defined as the total flow into :samp:`t`, the sink, is |
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384 stored in :samp:`R.graph['flow_value']`. Reachability to :samp:`t` using |
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385 only edges :samp:`(u, v)` such that |
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386 :samp:`R[u][v]['flow'] < R[u][v]['capacity']` induces a minimum |
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387 :samp:`s`-:samp:`t` cut. |
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388 |
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389 Examples |
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390 -------- |
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391 >>> from networkx.algorithms.flow import preflow_push |
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392 |
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393 The functions that implement flow algorithms and output a residual |
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394 network, such as this one, are not imported to the base NetworkX |
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395 namespace, so you have to explicitly import them from the flow package. |
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396 |
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397 >>> G = nx.DiGraph() |
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398 >>> G.add_edge("x", "a", capacity=3.0) |
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399 >>> G.add_edge("x", "b", capacity=1.0) |
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400 >>> G.add_edge("a", "c", capacity=3.0) |
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401 >>> G.add_edge("b", "c", capacity=5.0) |
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402 >>> G.add_edge("b", "d", capacity=4.0) |
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403 >>> G.add_edge("d", "e", capacity=2.0) |
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404 >>> G.add_edge("c", "y", capacity=2.0) |
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405 >>> G.add_edge("e", "y", capacity=3.0) |
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406 >>> R = preflow_push(G, "x", "y") |
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407 >>> flow_value = nx.maximum_flow_value(G, "x", "y") |
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408 >>> flow_value == R.graph["flow_value"] |
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409 True |
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410 >>> # preflow_push also stores the maximum flow value |
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411 >>> # in the excess attribute of the sink node t |
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412 >>> flow_value == R.nodes["y"]["excess"] |
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413 True |
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414 >>> # For some problems, you might only want to compute a |
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415 >>> # maximum preflow. |
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416 >>> R = preflow_push(G, "x", "y", value_only=True) |
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417 >>> flow_value == R.graph["flow_value"] |
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418 True |
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419 >>> flow_value == R.nodes["y"]["excess"] |
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420 True |
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421 |
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422 """ |
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423 R = preflow_push_impl(G, s, t, capacity, residual, global_relabel_freq, value_only) |
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424 R.graph["algorithm"] = "preflow_push" |
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425 return R |