Fixed database typo and removed unnecessary class identifier.
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venv/Lib/site-packages/networkx/algorithms/flow/mincost.py
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venv/Lib/site-packages/networkx/algorithms/flow/mincost.py
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"""
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Minimum cost flow algorithms on directed connected graphs.
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"""
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__all__ = ["min_cost_flow_cost", "min_cost_flow", "cost_of_flow", "max_flow_min_cost"]
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import networkx as nx
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def min_cost_flow_cost(G, demand="demand", capacity="capacity", weight="weight"):
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r"""Find the cost of a minimum cost flow satisfying all demands in digraph G.
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G is a digraph with edge costs and capacities and in which nodes
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have demand, i.e., they want to send or receive some amount of
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flow. A negative demand means that the node wants to send flow, a
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positive demand means that the node want to receive flow. A flow on
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the digraph G satisfies all demand if the net flow into each node
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is equal to the demand of that node.
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Parameters
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----------
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G : NetworkX graph
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DiGraph on which a minimum cost flow satisfying all demands is
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to be found.
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demand : string
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Nodes of the graph G are expected to have an attribute demand
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that indicates how much flow a node wants to send (negative
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demand) or receive (positive demand). Note that the sum of the
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demands should be 0 otherwise the problem in not feasible. If
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this attribute is not present, a node is considered to have 0
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demand. Default value: 'demand'.
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capacity : string
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Edges of the graph G are expected to have an attribute capacity
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that indicates how much flow the edge can support. If this
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attribute is not present, the edge is considered to have
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infinite capacity. Default value: 'capacity'.
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weight : string
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Edges of the graph G are expected to have an attribute weight
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that indicates the cost incurred by sending one unit of flow on
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that edge. If not present, the weight is considered to be 0.
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Default value: 'weight'.
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Returns
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-------
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flowCost : integer, float
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Cost of a minimum cost flow satisfying all demands.
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Raises
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------
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NetworkXError
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This exception is raised if the input graph is not directed or
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not connected.
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NetworkXUnfeasible
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This exception is raised in the following situations:
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* The sum of the demands is not zero. Then, there is no
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flow satisfying all demands.
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* There is no flow satisfying all demand.
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NetworkXUnbounded
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This exception is raised if the digraph G has a cycle of
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negative cost and infinite capacity. Then, the cost of a flow
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satisfying all demands is unbounded below.
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See also
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--------
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cost_of_flow, max_flow_min_cost, min_cost_flow, network_simplex
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Notes
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-----
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This algorithm is not guaranteed to work if edge weights or demands
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are floating point numbers (overflows and roundoff errors can
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cause problems). As a workaround you can use integer numbers by
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multiplying the relevant edge attributes by a convenient
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constant factor (eg 100).
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Examples
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--------
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A simple example of a min cost flow problem.
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>>> G = nx.DiGraph()
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>>> G.add_node("a", demand=-5)
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>>> G.add_node("d", demand=5)
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>>> G.add_edge("a", "b", weight=3, capacity=4)
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>>> G.add_edge("a", "c", weight=6, capacity=10)
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>>> G.add_edge("b", "d", weight=1, capacity=9)
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>>> G.add_edge("c", "d", weight=2, capacity=5)
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>>> flowCost = nx.min_cost_flow_cost(G)
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>>> flowCost
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24
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"""
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return nx.network_simplex(G, demand=demand, capacity=capacity, weight=weight)[0]
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def min_cost_flow(G, demand="demand", capacity="capacity", weight="weight"):
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r"""Returns a minimum cost flow satisfying all demands in digraph G.
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G is a digraph with edge costs and capacities and in which nodes
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have demand, i.e., they want to send or receive some amount of
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flow. A negative demand means that the node wants to send flow, a
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positive demand means that the node want to receive flow. A flow on
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the digraph G satisfies all demand if the net flow into each node
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is equal to the demand of that node.
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Parameters
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----------
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G : NetworkX graph
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DiGraph on which a minimum cost flow satisfying all demands is
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to be found.
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demand : string
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Nodes of the graph G are expected to have an attribute demand
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that indicates how much flow a node wants to send (negative
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demand) or receive (positive demand). Note that the sum of the
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demands should be 0 otherwise the problem in not feasible. If
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this attribute is not present, a node is considered to have 0
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demand. Default value: 'demand'.
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capacity : string
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Edges of the graph G are expected to have an attribute capacity
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that indicates how much flow the edge can support. If this
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attribute is not present, the edge is considered to have
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infinite capacity. Default value: 'capacity'.
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weight : string
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Edges of the graph G are expected to have an attribute weight
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that indicates the cost incurred by sending one unit of flow on
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that edge. If not present, the weight is considered to be 0.
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Default value: 'weight'.
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Returns
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-------
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flowDict : dictionary
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Dictionary of dictionaries keyed by nodes such that
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flowDict[u][v] is the flow edge (u, v).
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Raises
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------
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NetworkXError
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This exception is raised if the input graph is not directed or
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not connected.
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NetworkXUnfeasible
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This exception is raised in the following situations:
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* The sum of the demands is not zero. Then, there is no
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flow satisfying all demands.
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* There is no flow satisfying all demand.
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NetworkXUnbounded
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This exception is raised if the digraph G has a cycle of
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negative cost and infinite capacity. Then, the cost of a flow
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satisfying all demands is unbounded below.
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See also
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--------
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cost_of_flow, max_flow_min_cost, min_cost_flow_cost, network_simplex
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Notes
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-----
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This algorithm is not guaranteed to work if edge weights or demands
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are floating point numbers (overflows and roundoff errors can
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cause problems). As a workaround you can use integer numbers by
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multiplying the relevant edge attributes by a convenient
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constant factor (eg 100).
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Examples
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--------
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A simple example of a min cost flow problem.
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>>> G = nx.DiGraph()
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>>> G.add_node("a", demand=-5)
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>>> G.add_node("d", demand=5)
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>>> G.add_edge("a", "b", weight=3, capacity=4)
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>>> G.add_edge("a", "c", weight=6, capacity=10)
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>>> G.add_edge("b", "d", weight=1, capacity=9)
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>>> G.add_edge("c", "d", weight=2, capacity=5)
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>>> flowDict = nx.min_cost_flow(G)
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"""
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return nx.network_simplex(G, demand=demand, capacity=capacity, weight=weight)[1]
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def cost_of_flow(G, flowDict, weight="weight"):
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"""Compute the cost of the flow given by flowDict on graph G.
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Note that this function does not check for the validity of the
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flow flowDict. This function will fail if the graph G and the
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flow don't have the same edge set.
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Parameters
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----------
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G : NetworkX graph
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DiGraph on which a minimum cost flow satisfying all demands is
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to be found.
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weight : string
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Edges of the graph G are expected to have an attribute weight
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that indicates the cost incurred by sending one unit of flow on
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that edge. If not present, the weight is considered to be 0.
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Default value: 'weight'.
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flowDict : dictionary
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Dictionary of dictionaries keyed by nodes such that
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flowDict[u][v] is the flow edge (u, v).
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Returns
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-------
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cost : Integer, float
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The total cost of the flow. This is given by the sum over all
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edges of the product of the edge's flow and the edge's weight.
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See also
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--------
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max_flow_min_cost, min_cost_flow, min_cost_flow_cost, network_simplex
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Notes
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-----
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This algorithm is not guaranteed to work if edge weights or demands
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are floating point numbers (overflows and roundoff errors can
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cause problems). As a workaround you can use integer numbers by
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multiplying the relevant edge attributes by a convenient
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constant factor (eg 100).
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"""
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return sum((flowDict[u][v] * d.get(weight, 0) for u, v, d in G.edges(data=True)))
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def max_flow_min_cost(G, s, t, capacity="capacity", weight="weight"):
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"""Returns a maximum (s, t)-flow of minimum cost.
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G is a digraph with edge costs and capacities. There is a source
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node s and a sink node t. This function finds a maximum flow from
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s to t whose total cost is minimized.
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Parameters
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----------
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G : NetworkX graph
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DiGraph on which a minimum cost flow satisfying all demands is
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to be found.
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s: node label
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Source of the flow.
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t: node label
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Destination of the flow.
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capacity: string
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Edges of the graph G are expected to have an attribute capacity
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that indicates how much flow the edge can support. If this
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attribute is not present, the edge is considered to have
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infinite capacity. Default value: 'capacity'.
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weight: string
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Edges of the graph G are expected to have an attribute weight
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that indicates the cost incurred by sending one unit of flow on
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that edge. If not present, the weight is considered to be 0.
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Default value: 'weight'.
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Returns
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-------
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flowDict: dictionary
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Dictionary of dictionaries keyed by nodes such that
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flowDict[u][v] is the flow edge (u, v).
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Raises
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------
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NetworkXError
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This exception is raised if the input graph is not directed or
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not connected.
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NetworkXUnbounded
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This exception is raised if there is an infinite capacity path
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from s to t in G. In this case there is no maximum flow. This
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exception is also raised if the digraph G has a cycle of
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negative cost and infinite capacity. Then, the cost of a flow
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is unbounded below.
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See also
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--------
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cost_of_flow, min_cost_flow, min_cost_flow_cost, network_simplex
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Notes
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-----
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This algorithm is not guaranteed to work if edge weights or demands
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are floating point numbers (overflows and roundoff errors can
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cause problems). As a workaround you can use integer numbers by
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multiplying the relevant edge attributes by a convenient
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constant factor (eg 100).
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Examples
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--------
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>>> G = nx.DiGraph()
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>>> G.add_edges_from(
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... [
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... (1, 2, {"capacity": 12, "weight": 4}),
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... (1, 3, {"capacity": 20, "weight": 6}),
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... (2, 3, {"capacity": 6, "weight": -3}),
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... (2, 6, {"capacity": 14, "weight": 1}),
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... (3, 4, {"weight": 9}),
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... (3, 5, {"capacity": 10, "weight": 5}),
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... (4, 2, {"capacity": 19, "weight": 13}),
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... (4, 5, {"capacity": 4, "weight": 0}),
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... (5, 7, {"capacity": 28, "weight": 2}),
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... (6, 5, {"capacity": 11, "weight": 1}),
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... (6, 7, {"weight": 8}),
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... (7, 4, {"capacity": 6, "weight": 6}),
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... ]
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... )
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>>> mincostFlow = nx.max_flow_min_cost(G, 1, 7)
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>>> mincost = nx.cost_of_flow(G, mincostFlow)
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>>> mincost
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373
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>>> from networkx.algorithms.flow import maximum_flow
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>>> maxFlow = maximum_flow(G, 1, 7)[1]
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>>> nx.cost_of_flow(G, maxFlow) >= mincost
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True
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>>> mincostFlowValue = sum((mincostFlow[u][7] for u in G.predecessors(7))) - sum(
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... (mincostFlow[7][v] for v in G.successors(7))
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... )
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>>> mincostFlowValue == nx.maximum_flow_value(G, 1, 7)
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True
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"""
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maxFlow = nx.maximum_flow_value(G, s, t, capacity=capacity)
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H = nx.DiGraph(G)
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H.add_node(s, demand=-maxFlow)
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H.add_node(t, demand=maxFlow)
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return min_cost_flow(H, capacity=capacity, weight=weight)
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