Data Structures & Algorithms
Network Flow
Richard Newmanbased on book by R. Sedgewick
Network Flow Problems
• Weighted, digraph G, or network• May have cost per unit flow for edges• May have maximum flow per edge• May have max production rates• May have required consumption rate
per sink
Distribution Problems• Merchandise distribution
– Sources with production rates– Sinks with consumption rates– Distribution centers
• Input rate = output rate– Channels with maximum rate and
unit cost for distribution
Merchandise distribution
Get product to retail locations cheaply
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RetailLocations
Factories
DistributionCenters
Transportation Problems• Communications
– Max total data rate between a source and sink
– Cheapest way to move a given amount of data from s to t
• Traffic flow– Minimize evacuation time– Minimize total cost
Transportation Problem
No channel capacity restrictionsGet product to retail locations cheaply
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Demand
Supply
Channels
Matching Problems• Job placement
– Interviews + job offers– Maximize number of placements
• Min-distance point matching– Two sets A and B of N points each– Find set of N segments matching
an element from A with one from B that has lowest cost
Matching Problem
Maximize placements (matching)
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Applicant
Employer
Offers
Cut Problems• Network reliability
– What is minimum number of lines that must be cut to disconnect two switches?
• Supply line cutting– What is the minimum supply line
destruction required to ensure no troops get supplies?
Cut Problem
How few edges must be cut to disrupt deliveryMay have edge weights also
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Troops
Supplies
Delivery paths
Network Flow Problems• Generic problems• Maxflow
– What is maximum flow between s and t?
• Mincost-flow– What is cheapest cost way to
achieve a particular flow?
Network Flow
• Flow Networks• Maxflow Algorithms• Maxflow Reductions• Mincost Flows• Network Simplex Algorithm
Flow Networks• Defn 22.1: A network with a single
source and a single sink is an s-t network
Flow Networks• Defn 22.2: A flow network is an s-t
network with positive edge weights, called capacities.
• A flow in a flow network is a set of non-negative edge weights called edge flows satisfying:
• No edge flow exceeds capacity• In flow = out flow for interior nodes
Flow Network
Maximize flow subject to capacityand conservation of flow
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Pipelines and Valves
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Is this OK?
Flow Network
Can we do better?
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Pipelines and Valves
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Flow Network
Are we done?Is this OK?
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Flow Network
Now are we done?Yep
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Flow Network • Sum of flows into a node is called
inflow• Sum of flows out of a node is called
outflow• Conservation of flow: except for
source and sink, inflow = outflow• Feasible flow = obeys constraints
(max flow and conservation of flow)
Flow Network • Set outflow from sink to zero• Set inflow to source to zero• Outflow of source = inflow of sink• This is called network's value
Maximum Flow • Given an s-t network, find a flow
such that no other flow from s to t has a larger value.
• A flow like this is called a maxflow.• Problem of finding one is called the
maxflow problem.
Augmenting Path Maximum Flow • First algorithm like this due to Ford
and Fulkerson • Iteratively:
• Find a feasible path from s to t• Find the max residual capacity on it• Saturate the path by adding flow
along it equal to the minimum residual capacity
Ford-Fulkerson
Find path with capacity left: 0-1-3-5Find minimum residual capacity: Add flow to saturate: 2
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New flow
Ford-Fulkerson
Find path with capacity left: 0-2-4-5Find minimum residual capacity: Add flow to saturate: 1
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Saturated edge, flow
Unsaturated flow
Ford-Fulkerson
Find path with capacity left: Wait a doggone second... Consider flow in one direction as
capacity in reverse direction!
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There are none!
“Reverse edge”
Ford-Fulkerson
What is “residual capacity” of 3-1? Find path with capacity left: What is the minimum RC on this path?Add flow to saturate...
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0-2-3-1-4-52 (the flow)
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“Reverse edge”
Ford-Fulkerson
Find path with capacity left: We are done.
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There is none.
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Ford-Fulkerson
Removing saturated edges partitions the network
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Ford-Fulkerson
Removing saturated edges partitions the network
Source s is in one part, t is in anotherNote: cut is subset of saturated edges
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s-t Cut
• Defn. 22.3: An s-t cut is a cut that places node s in one of its sets and node t in the other. Its capacity is the sum of its edge weights.
The flow across an s-t cut is the sum of the flows across its s-t edges, less the sum of the flows across its t-s edges.
MinCut Problem
• Given an s-t network, find an s-t cut such that the capacity of no other cut is smaller. We call this a mincut.
• Maxflow-mincut theorem – min cut capacity and maxflow value are equal
Residual Network• Given flow network G and a flow F,
the residual network R for the flow has the same nodes, and one or two edges in R for each edge in the original: for edge (v,w) in G, let f be the flow and c be the capacity. If f<c, include (v,w) with capacity c-f; if>0 include (w,v) with capacity f.
Residual Network• If flow is 0, then the original edge
with original capacity is used• For positive flows, decrease capacity
by the flow (unless the residual capacity is zero – the edge is saturated – remove the edge)
• For positive flows, add a reverse directed edge with the flow as its capacity (can reduce flow that much).
Flow Network with Flow
Make reverse edges with capacity = flowDecrease capacity by flow on forward edges
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Flow
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2 Reverseedges
Residual Network
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Saturated edge
Residual capacity
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Make reverse edges with capacity = flowDecrease capacity by flow on forward edgesRemove edges with zero residual capacity(saturated edges)
Residual Network
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1 Residual capacity
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2 Reverseedges
Residual network remainsNote that reverse edge to s or from t don’t
help, so can be omitted in practice
Variants on Ford-Fulkerson• Shortest Augmenting Path
• Measured by number of edges• Build “layer graph” – like BFS• Which nodes can be reached by 1
edge, then by 2 edges, etc.• Stop when t is reached and add
flow to saturate edge on path to t
Variants on Ford-Fulkerson• Maximum Flow Augmenting Path
• Measured by max flow along path• Pick edges that give max flow to
next layer• Take max of (min flow along path to
predecessor u, capacity (u,v)) to find max flow to v through u at that layer
Ford-Fulkerson Complexity• Prop. 22.6: Let M be the maximum
edge capacity in G. The number of augmenting paths needed by any implementation of F-F is at most VM.
• Every AP adds at least one unit of flow to every cut; any cut has at most V edges; hence the algorithm must terminate after VM passes since any cut must be saturated by then.
Ford-Fulkerson Complexity• Cor: The time required to find a
maxflow is O(VEM), which is O(V2M) for sparse networks.
• Linear (in edges) graph search per pass.
• Need extra lg V factor if using priority queue fringe implementation
• Actual performance is quite good
Ford-Fulkerson Complexity• Prop. 22.7: The number of
augmenting paths needed in the shortest augmenting path F-F algo is at most VE/2.
Ford-Fulkerson Complexity• The length of the APs monotone non-
decreasing. • Every AP has a critical edge that is
saturated in its pass. • Each time edge e is the critical edge,
the AP must be at least 2 hops longer.
• The longest path has < V edges
Ford-Fulkerson Complexity• Prop. 22.7 (again): The number of
augmenting paths needed in the shortest augmenting path F-F algo is at most VE/2.
• Cor: The time required to find a maxflow in a sparse network is O(V3)
• O(E) time per pass, VE/2 passes, and if G is sparse, E is O(V).
Ford-Fulkerson Complexity• Prop. 22.8: The number of
augmenting paths needed in the maximal augmenting path F-F algo is at most 2E lg M.
• Cor: The time required to find a maxflow in a sparse network is O(V2 lg M lg V)
Network Flow
• Flow Networks• Maxflow Algorithms• Maxflow Reductions• Mincost Flows• Network Simplex Algorithm
Maxflow in General Networks• Maximize the total outflow from
sources in a network. (Zero by convention if no sources or sinks)
• Multiple sources• Multiple sinks• Still need feasible flows!
Maxflow in General Networks• Prop. 22.14: The maxflow in general
networks is equivalent to the maxflow problem for general networks.
• The general case subsumes the special case of s-t networks.
• Add dummy source connected to all sources and dummy sink to all sinks by high-capacity edges
Vertex-capacity Constraints• Given a flow network, find a maxflow
satisfying additional constraints that the flow through each node v must not exceed the capacity of that node.
Vertex-capacity Constraints• Prop. 22.15: The maxflow with node
capacity problem is equivalent to maxflow problem.
• Setting node capacity high subsumes the special case of s-t networks.
• Node capacity can be set to max of in-capacity and out-capacity (sum of in- and out-edge capacities, resp.)
Vertex-capacity Constraints• Prop. 22.15: The maxflow with node
capacity problem is equivalent to maxflow problem.
• Split each node into in-part and out-part, with all in-edge to in-part and all out-edge from out-part
• Make edge from in-part to out-part with edge capacity = node capacity in original problem
Maxflow in Acyclic Networks• Maximize the total outflow from
sources in an acyclic network. • Seems like it might be easier• But NO!• Easy transformation from any
network to acyclic network• V nodes -> 2V + 2 nodes• E edges -> E + 3V edges
Maxflow in General Networks• Prop. 22.16: The maxflow in acyclic
networks is equivalent to the maxflow problem for general networks.
• The construction allows a mincut in original to reveal a mincut in the transformed version, and vice versa, with transformed cut values. p427
Maxflow in Undirected Networks• Prop. 22.17: The maxflow in
undirected networks reduces to the maxflow problem for s-t networks.
• Just make two directed edges in the s-t network for every edge in the undirected network
• Make both edge capacities equal to the corresponding edge capacity in the undirected network
Maxflow in Undirected Networks• Note that we did not say equivalent!• It remains possible that finding
maxflows in undirected networks is easier than finding them in directed networks.
Feasible Flow• Given a flow network G, assign a
weight to each node, interpreted as supply if positive, and demand if negative.
• A flow is feasible if it obeys flow constraints AND if the difference between the outflow sum and inflow sum for each node v = v’s weight
• Finding a feasible flow is FF problem
Feasible Flow• Prop. 22.18: The feasible flow problem
reduces to maxflow• Make flow network with V more nodes.• For supply nodes, make the new node
a source with edge weight the same as the node weight
• For demand nodes make the new node a sink with edge capacity = -weight
Feasible Flow• Prop. 22.18: The feasible flow problem
reduces to maxflow• Feasible flow exists in original network
iff maxflow in constructed flow network saturates all edges between sources and companions and all edges between sinks and companions
Max-cardinality Bipartite Matching• Given a bipartite graph G, find a set of
edges of maximum cardinality such that each vertex is connected to at most one other vertex.
• A.k.a. Bipartite Matching Problem• Direct solution by trying all possibilities
results in combinatorial explosion!
Max-cardinality Bipartite Matching• Prop. 22.19: Bipartite matching reduces
to the maxflow problem.• Make edges directed from set A to set
B nodes• Add two dummy nodes s and t• Add edges from s to each node in A• Add edges from each node in B to t• Assign capacity 1 to all edges
Edge-Connectivity• Prop. 22.20: The minimum number of
edges whose removal disconnects two nodes in a digraph is equal to the number of edge-disjoint paths between the two nodes.
• Define a flow network with the same nodes and edges, with capacities all 1.
• The capacity of any s-t cut is equal to the cut’s cardinality
Connectivity Problems• Edge connectivity• Node connectivity• Digraphs• Undirected graphs• Hence 4 problems• Can do reductions….
Edge-Connectivity• Prop. 22.21: The time required to
determine the edge-connectivity of an undirected graph G is O(E2)
• Compute min cut in corresponding s-t network with unit capacity edges.
• Edge connectivity for G is the minimum over all s-t pairs
• Note we only have to use a minimal degree vertex as s – faster
Network Flow
• Flow Networks• Maxflow Algorithms• Maxflow Reductions• Mincost Flows• Network Simplex Algorithm
Mincost Flows• Flow network has cost also associated
with each edge• Defn 22.8: Flow cost of an edge is the
product of the flow and the edge cost. The cost of a flow is the sum of the flow costs of that flow’s edges.
Mincost Flows• Mincost Maxflow
• Find maxflow with minimum flow cost• Mincost Feasible Flow
• Find feasible flow with minimum cost• Prop 22.22: The mincost feasible flow
and the mincost maxflow problems are equivalent.
Network Flow
• Flow Networks• Maxflow Algorithms• Maxflow Reductions• Mincost Flows• Network Simplex Algorithm