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Ford-Fulkerson Algorithm
Ford-Fulkerson Algorithm
Overview
� Network flows on directed acyclic graphs
� Ford-Fulkerson Algorithm
-Residual networks
CS 4407, Algorithms , University College Cork,, Gregory M. Provan
Network flows on directed acyclic graphsNetwork flows on directed acyclic graphs
� DAG is a directed graph with no cycles.
� Flow network:– Flow network G = (V,E) is a directed graph in which each edge (u,v) ∈∈∈∈ EEEE
hashashashas aaaa nonnegativenonnegativenonnegativenonnegative capacitycapacitycapacitycapacity c(c(c(c(u,vu,vu,vu,v)))) >=>=>=>= 0000....– IfIfIfIf ((((u,vu,vu,vu,v)))) ∉∉∉∉ E , we assume that c(u,v) = 0.
� Distinguished vertices: source s,s, and sink tt.– Source: produces the material at a steady rate .
CS 4407, Algorithms , University College Cork,, Gregory M. Provan
– Source: produces the material at a steady rate .
– Sink: consumes the material at a steady rate.
� Objective : compute how much material can flow through thenetwork
Flow: A flow in G is a real-valued function f : V × V → R that satisfies three properties:
� 1. Capacity constraint : For all u,v ∈∈∈∈ V, we require f(u,v) ≤ c(u,v).– The net flow from one vertex to another must not exceed the given capacity.
� 2. Skew symmetry : For all u,v ∈ ∈ ∈ ∈ V, we require f(u,v) = -f(v,u).
� The net flow from a vertex u to a vertex v is the negative of the net flow in the other direction
� 3. Flow conservation : For all u ∈∈∈∈ V - {s,t}, we require
Network Network PropertiesProperties
CS 4407, Algorithms , University College Cork,, Gregory M. Provan
� 3. Flow conservation : For all u ∈∈∈∈ V - {s,t}, we require
∑ f(u,v) = 0.
v ∈ ∈ ∈ ∈ V
The quantity f(u,v),which can be positive or negative , is called the net flow from vertex u to v.
The value of the flow is defined as | f | = ∑ f(s,v) that is the total net
v ∈∈∈∈ V
Flow out of the source.
Residual Capacity
� Given a flow f in network G = (V, E).
� Consider a pair of vertices u, v ∈∈∈∈ V.
� Residual capacity: – The amount of additional flow we can push directly from u to v.
– cf (u, v) = c(u, v) −−−− f (u, v)
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CS 4407, Algorithms , University College Cork,, Gregory M. Provan
– cf (u, v) = c(u, v) −−−− f (u, v)≥ 0 (since f (u, v) ≤ c(u, v))
� Example:– c(u,v) = 16, f(u,v) = 5 ���� cf (u, v) = 11
Residual Network
� Residual network: Gf = (V, Ef ),
Ef = {(u, v) ∈∈∈∈ V × V : cf (u, v) > 0} .
� Each edge of the residual network can admit a positive flow.
� Given flows f1 and f2, the flow sum
CS 4407, Algorithms , University College Cork,, Gregory M. Provan
� Given flows f1 and f2, the flow sum f1 + f2 is
– (f1 + f2)(u, v) = f1(u, v) + f2(u, v) for all u, v ∈∈∈∈ V.
Flow Networks: Simple Example
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CS 4407, Algorithms , University College Cork,, Gregory M. Provan
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Example p.2
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Step 2: path {s,a,t}
CS 4407, Algorithms , University College Cork,, Gregory M. Provan
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Step 1: path {s,a,b,t}Step 2: path {s,a,t}
Step 3:Path {s,b,t}
Example: p.3
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CS 4407, Algorithms , University College Cork,, Gregory M. Provan
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Step 3 Step 4
Example: Max-Flow Min-Cut
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CS 4407, Algorithms , University College Cork,, Gregory M. Provan
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Ford-Fulkerson Algorithm
� The Ford_Fulkerson method is iterative,
� Starts with f(u,v) for (u,v) ∈∈∈∈ V, initial flow of value 0.
� The method is based on the augmenting path
CS 4407, Algorithms , University College Cork,, Gregory M. Provan
� The method is based on the augmenting path– a path from s to t along which we can push more flow and
then augment flow along this path.
Ford-Fulkerson Algorithm
Ford-Fulkerson(G, s, t) ; G = (V, E) 1 for each edge (u,v) in E 2 f(u,v) = f(v,u) = 0 3 while ∃∃∃∃ path p from s to t in residual network Gf4 do cf (p) = min{cf (u, v) : (u, v) is in p}
5 for each edge (u,v) on p
CS 4407, Algorithms , University College Cork,, Gregory M. Provan
5 for each edge (u,v) on p 6 f(u,v) = f(u,v) + cf (p) 7 f(v,u) = -f(u,v)
� Running time for this algorithms is O(E * |f*|) where f* is the maximum flow found by algorithm.
� First three Lines take time θ(E).
Complexity: Ford-Fulkerson Algorithm
CS 4407, Algorithms , University College Cork,, Gregory M. Provan
� First three Lines take time θ(E).
� The while loop of last five lines is executed at most |f*| times, since the flow value increases by at least one unit in each iteration.
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Example:
CS 4407, Algorithms , University College Cork,, Gregory M. Provan
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CS 4407, Algorithms , University College Cork,, Gregory M. Provan
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CS 4407, Algorithms , University College Cork,, Gregory M. Provan
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CS 4407, Algorithms , University College Cork,, Gregory M. Provan
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CS 4407, Algorithms , University College Cork,, Gregory M. Provan
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Reference
� Class Notes from Dr. Istvan Jonyer of
Oklahoma State University
� Class Notes from UTexas CSE 5311 of 2004,
made by Hiren Patel, Ujjval Patel
CS 4407, Algorithms , University College Cork,, Gregory M. Provan
Backup: simple network
CS 4407, Algorithms , University College Cork,, Gregory M. Provan
Consider the network G=(V,E) shown in the figure below. Each
edge (u,v) ∈∈∈∈ E in the network is labeled with its capacity c(u,v).
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CS 4407, Algorithms , University College Cork,, Gregory M. Provan
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Flow: 1
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Residual graph
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CS 4407, Algorithms , University College Cork,, Gregory M. Provan
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CS 4407, Algorithms , University College Cork,, Gregory M. Provan
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CS 4407, Algorithms , University College Cork,, Gregory M. Provan
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CS 4407, Algorithms , University College Cork,, Gregory M. Provan
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Max Flow : 5
