183 lines
5.6 KiB
Python
183 lines
5.6 KiB
Python
"""
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Utility classes and functions for network flow algorithms.
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"""
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from collections import deque
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import networkx as nx
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__all__ = [
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"CurrentEdge",
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"Level",
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"GlobalRelabelThreshold",
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"build_residual_network",
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"detect_unboundedness",
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"build_flow_dict",
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]
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class CurrentEdge:
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"""Mechanism for iterating over out-edges incident to a node in a circular
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manner. StopIteration exception is raised when wraparound occurs.
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"""
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__slots__ = ("_edges", "_it", "_curr")
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def __init__(self, edges):
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self._edges = edges
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if self._edges:
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self._rewind()
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def get(self):
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return self._curr
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def move_to_next(self):
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try:
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self._curr = next(self._it)
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except StopIteration:
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self._rewind()
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raise
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def _rewind(self):
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self._it = iter(self._edges.items())
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self._curr = next(self._it)
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class Level:
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"""Active and inactive nodes in a level.
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"""
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__slots__ = ("active", "inactive")
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def __init__(self):
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self.active = set()
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self.inactive = set()
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class GlobalRelabelThreshold:
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"""Measurement of work before the global relabeling heuristic should be
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applied.
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"""
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def __init__(self, n, m, freq):
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self._threshold = (n + m) / freq if freq else float("inf")
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self._work = 0
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def add_work(self, work):
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self._work += work
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def is_reached(self):
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return self._work >= self._threshold
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def clear_work(self):
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self._work = 0
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def build_residual_network(G, capacity):
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"""Build a residual network and initialize a zero flow.
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The residual network :samp:`R` from an input graph :samp:`G` has the
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same nodes as :samp:`G`. :samp:`R` is a DiGraph that contains a pair
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of edges :samp:`(u, v)` and :samp:`(v, u)` iff :samp:`(u, v)` is not a
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self-loop, and at least one of :samp:`(u, v)` and :samp:`(v, u)` exists
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in :samp:`G`.
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For each edge :samp:`(u, v)` in :samp:`R`, :samp:`R[u][v]['capacity']`
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is equal to the capacity of :samp:`(u, v)` in :samp:`G` if it exists
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in :samp:`G` or zero otherwise. If the capacity is infinite,
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:samp:`R[u][v]['capacity']` will have a high arbitrary finite value
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that does not affect the solution of the problem. This value is stored in
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:samp:`R.graph['inf']`. For each edge :samp:`(u, v)` in :samp:`R`,
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:samp:`R[u][v]['flow']` represents the flow function of :samp:`(u, v)` and
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satisfies :samp:`R[u][v]['flow'] == -R[v][u]['flow']`.
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The flow value, defined as the total flow into :samp:`t`, the sink, is
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stored in :samp:`R.graph['flow_value']`. If :samp:`cutoff` is not
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specified, reachability to :samp:`t` using only edges :samp:`(u, v)` such
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that :samp:`R[u][v]['flow'] < R[u][v]['capacity']` induces a minimum
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:samp:`s`-:samp:`t` cut.
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"""
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if G.is_multigraph():
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raise nx.NetworkXError("MultiGraph and MultiDiGraph not supported (yet).")
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R = nx.DiGraph()
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R.add_nodes_from(G)
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inf = float("inf")
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# Extract edges with positive capacities. Self loops excluded.
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edge_list = [
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(u, v, attr)
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for u, v, attr in G.edges(data=True)
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if u != v and attr.get(capacity, inf) > 0
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]
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# Simulate infinity with three times the sum of the finite edge capacities
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# or any positive value if the sum is zero. This allows the
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# infinite-capacity edges to be distinguished for unboundedness detection
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# and directly participate in residual capacity calculation. If the maximum
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# flow is finite, these edges cannot appear in the minimum cut and thus
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# guarantee correctness. Since the residual capacity of an
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# infinite-capacity edge is always at least 2/3 of inf, while that of an
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# finite-capacity edge is at most 1/3 of inf, if an operation moves more
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# than 1/3 of inf units of flow to t, there must be an infinite-capacity
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# s-t path in G.
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inf = (
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3
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* sum(
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attr[capacity]
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for u, v, attr in edge_list
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if capacity in attr and attr[capacity] != inf
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)
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or 1
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)
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if G.is_directed():
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for u, v, attr in edge_list:
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r = min(attr.get(capacity, inf), inf)
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if not R.has_edge(u, v):
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# Both (u, v) and (v, u) must be present in the residual
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# network.
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R.add_edge(u, v, capacity=r)
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R.add_edge(v, u, capacity=0)
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else:
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# The edge (u, v) was added when (v, u) was visited.
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R[u][v]["capacity"] = r
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else:
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for u, v, attr in edge_list:
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# Add a pair of edges with equal residual capacities.
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r = min(attr.get(capacity, inf), inf)
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R.add_edge(u, v, capacity=r)
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R.add_edge(v, u, capacity=r)
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# Record the value simulating infinity.
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R.graph["inf"] = inf
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return R
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def detect_unboundedness(R, s, t):
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"""Detect an infinite-capacity s-t path in R.
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"""
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q = deque([s])
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seen = {s}
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inf = R.graph["inf"]
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while q:
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u = q.popleft()
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for v, attr in R[u].items():
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if attr["capacity"] == inf and v not in seen:
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if v == t:
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raise nx.NetworkXUnbounded(
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"Infinite capacity path, flow unbounded above."
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)
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seen.add(v)
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q.append(v)
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def build_flow_dict(G, R):
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"""Build a flow dictionary from a residual network.
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"""
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flow_dict = {}
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for u in G:
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flow_dict[u] = {v: 0 for v in G[u]}
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flow_dict[u].update(
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(v, attr["flow"]) for v, attr in R[u].items() if attr["flow"] > 0
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)
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return flow_dict
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