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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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