Data Structures &
Algorithms
Network Flow
Richard Newman
based on book by R. Sedgewick
Network Flow Problems
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•
•
•
•
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
0
7
1
72
3
4
5
6
8
Factories
Distribution
Centers
Retail
9 Locations
Get product to retail locations cheaply
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
0
1
72
3
4
Supply
Channels
5
6
7
8
9
Demand
No channel capacity restrictions
Get product to retail locations cheaply
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
0
1
72
3
4
Employer
Offers
5
6
7
8
9
Maximize placements (matching)
Applicant
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
0
1
72
3
4
5
6
Supplies
Delivery
paths
Troops
7
8
9
How few edges must be cut to disrupt delivery
May have edge weights also
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
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•
•
•
•
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
3
1
2
0
1
3
Oil 3
Field
3
5
1
1
2
12
11
1
4
3 Refinery
Pipelines
and Valves
Maximize flow subject to capacity
Is this OK?
and conservation of flow
Flow Network
1
1
0
2
3
1
2
Oil 3
Field
1 3
5
1
1
2
12
11
1
4
Pipelines
and Valves
Can we do better?
3 Refinery
Flow Network
1
1
0
2
3
1 1
2
Oil 3
Field
1 3
5
1
1
2
22
11
1
4
Pipelines
and Valves
Are we done?
Is this OK?
3 Refinery
Flow Network
1
2
0
2
3
1 1
2
Oil 3
Field
1 3
5
1
1
2
22
11
2
4
Pipelines
and Valves
Now are we done?
Yep
3 Refinery
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
1
2
2
2 3
3 2
1
2
s 0
5 t
1
3
2
1
3
New flow
4
Find path with capacity left: 0-1-3-5
Find minimum residual capacity: 2
Add flow to saturate: 2
Ford-Fulkerson
1
2
s 0
2
2 3
3
1
1
1
3
2
1 1
1
4
Saturated
22
edge,
flow
5 t
3
Unsaturated
flow
Find path with capacity left: 0-2-4-5
Find minimum residual capacity: 1
Add flow to saturate: 1
Ford-Fulkerson
1
2
s 0
2
2 3
3
1
22
5 t
1
1
3
2
11
1
4
3
“Reverse
edge”
Find path with capacity left: There are none!
Wait a doggone second...
Consider flow in one direction as
capacity in reverse direction!
Ford-Fulkerson
2
s 0
2
2 3
1
2
1
3
22
5 t
1
3
1
1
4
3
“Reverse
edge”
11
2
1
What is “residual capacity” of 3-1? 2 (the flow)
Find path with capacity left: 0-2-3-1-4-5
What is the minimum RC on this path? 1
Add flow to saturate...
Ford-Fulkerson
2
s 0
2
2
3
1 3
1
1 1
1
1
1
3
1
22
5 t
1
2
4
3
11
2
1
Find path with capacity left: There is none.
We are done.
Ford-Fulkerson
2
s 0
2
2
3
1 3
1
1 1
1
1
1
3
1
22
5 t
1
2
4
3
11
2
1
Removing saturated edges partitions
the network
Ford-Fulkerson
1
s 0
3
3
5 t
2
3
2
2
4
3
Removing saturated edges partitions
the network
Source s is in one part, t is in another
Note: cut is subset of saturated edges
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
2 3
1
2
2
1
2 2
3 2
2
2
Reverse
5 t
edges
s 0
1
3
2
1
3
Flow
4
Make reverse edges with capacity = flow
Decrease capacity by flow on forward edges
Residual Network
1
2
Saturated
edge
0
0
1
2
2
Reverse
s 0
5 t
edges
1
3
3
Residual
1
2
4
capacity
Make reverse edges with capacity = flow
Decrease capacity by flow on forward edges
Remove edges with zero residual capacity
(saturated edges)
1
3
Residual Network
1
2
1
3
1
2
2
Reverse
5 t
edges
s 0
1
3
2
1
3
4
Residual
capacity
Residual network remains
Note 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 nondecreasing.
• 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
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•
•
•
•
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 outpart, 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
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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
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•
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•
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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
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•
•
•
•
Flow Networks
Maxflow Algorithms
Maxflow Reductions
Mincost Flows
Network Simplex Algorithm