Ford Fulkerson Algorithm

Information about Ford Fulkerson Algorithm

The Ford-Fulkerson algorithm (named for L. R. Ford, Jr. and D. R. Fulkerson) computes the maximum flow in a flow network. It was published in 1956. The name Ford-Fulkerson is often also used for the Edmonds-Karp algorithm, which is a specialisation of Ford-Fulkerson.

The idea behind the algorithm is very simple: As long as there is a path from the source (start node) to the sink (end node), with available capacity on all edges in the path, we send flow along one of these paths. Then we find another path, and so on. A path with available capacity is called an augmenting path.

Algorithm

Given is a graph $\text{[math]}$ , with capacity $\text{[math]}$ and flow $\text{[math]}$ for the edge from $\text{[math]}$ to $\text{[math]}$ . We want to find the maximum flow from the source $\text{[math]}$ to the sink $\text{[math]}$ . After every step in the algorithm the following is maintained:

$\text{[math]}$ . The flow from $\text{[math]}$ to $\text{[math]}$ does not exceed the capacity.
$\text{[math]}$ . Maintain the net flow between $\text{[math]}$ and $\text{[math]}$ . If in reality $\text{[math]}$ units are going from $\text{[math]}$ to $\text{[math]}$ , and $\text{[math]}$ units from $\text{[math]}$ to $\text{[math]}$ , maintain $\text{[math]}$ and $\text{[math]}$ .
$\text{[math]}$ for all nodes $\text{[math]}$ , except $\text{[math]}$ and $\text{[math]}$ . The amount of flow into a node equals the flow out of the node.

This means that the flow through the network is a legal flow after each round in the algorithm. We define the residual network $\text{[math]}$ to be the network with capacity $\text{[math]}$ and no flow. Notice that it is not certain that $\text{[math]}$ , as sending flow on $\text{[math]}$ might close $\text{[math]}$ (it is saturated), but open a new edge $\text{[math]}$ in the residual network.

Algorithm Ford-Fulkerson

Inputs Graph $\text{[math]}$ with flow capacity $\text{[math]}$ , a source node $\text{[math]}$ , and a sink node $\text{[math]}$

Output A flow $\text{[math]}$ from $\text{[math]}$ to $\text{[math]}$ which is a maximum

# $\text{[math]}$ for all edges $\text{[math]}$

# While there is a path $\text{[math]}$ from $\text{[math]}$ to $\text{[math]}$ in $\text{[math]}$ , such that $\text{[math]}$ for all edges $\text{[math]}$ :

## Find $\text{[math]}$

## For each edge $\text{[math]}$

### $\text{[math]}$ (Send flow along the path)

### $\text{[math]}$ (The flow might be "returned" later)

The path can be found with for example a breadth-first search or a depth-first search in $\text{[math]}$ . If you use the former, the algorithm is called Edmonds-Karp.

Complexity

By adding the flow augmenting path to the flow already established in the graph, the maximum flow will be reached when no more flow augmenting paths can be found in the graph. However, there is no certainty that this situation will ever be reached, so the best that can be guaranteed is that the answer will be correct if the algorithm terminates. In the case that the algorithm runs forever, the flow might not even converge towards the maximum flow. However, this situation only occurs with irrational flow values. When the capacities are integers, the runtime of Ford-Fulkerson is bounded by O(E*f), where E is the number of edges in the graph and f is the maximum flow in the graph. This is because each augmenting path can be found in O(E) time and increases the flow by an integer amount which is at least 1.

A variation of the Ford-Fulkerson algorithm with guaranteed termination and a runtime independent of the maximum flow value is the Edmonds-Karp algorithm, which runs in O(VE²) time.

Example

The following example show the first steps of Ford-Fulkerson in a flow network with 4 nodes, source A and sink D. The augmenting paths are found with a depth-first-search, where neighbours are visited in alphabetical order. This example shows the worst-case behaviour of the algorithm. In each step, only a flow of 1 is sent across the network. See that if you used a breadth-first-search instead, you would only need two steps.

Path	Capacity	Resulting flow network
	Initial flow network
$\text{[math]}$	$\text{[math]}$ $\text{[math]}$ $\text{[math]}$
$\text{[math]}$	$\text{[math]}$ $\text{[math]}$ $\text{[math]}$
	After 1998 more steps $\text{[math]}$
	Final flow network

Notice how flow is "pushed back" from C to B when finding the path A,C,B,D.

External links

Animation of Ford-Fulkerson algorithm

References

Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 26.2: The Ford-Fulkerson method, pp.651–664.

Lester Randolph Ford, Jr. (born September 23, 1927) is an American mathematician specializing in network flow programming, and son of Lester R. Ford, Sr..

His 1956 paper with D. R. Fulkerson on the maximum flow problem established the maxflow-mincut theorem.
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Delbert Ray Fulkerson (August 14, 1924 - January 10, 1976) was a mathematician who co-developed the Ford-Fulkerson algorithm, one of the most used algorithms to compute maximal flows in networks.

Fulkerson received his Ph.D. at the University of Wisconsin-Madison in 1951.
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maximum flow problem is to find a feasible flow through a single-source, single-sink flow network that is maximum ^[1]. Sometimes it is defined as finding the value of such a flow.
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In graph theory, a flow network is a directed graph where each edge has a capacity and each edge receives a flow. The amount of flow on an edge may not exceed the capacity of the edge.
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In computer science and graph theory, the Edmonds-Karp algorithm is an implementation of the Ford-Fulkerson method for computing the maximum flow in a flow network in . It is asymptomatically slower than the relabel-to-front algorithm, which runs in , but it is often faster in
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In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will proceed through a well-defined series of successive states, eventually terminating in an
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General Data

Class: Search Algorithm
Data Structure: Graph
Time Complexity:
Space Complexity:
Optimal: yes (for unweighted graphs)
Complete: yes

In graph theory, breadth-first search (
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General Data

Class: Search algorithm
Data Structure: Graph
Time Complexity:
Space Complexity: where is the length of the longest simple path in the graph.
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In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithm's usage of computational resources (usually running time or memory).
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Thomas H. Cormen is the co-author of Introduction to Algorithms, along with Charles Leiserson, Ron Rivest, and Cliff Stein. He is a Full Professor of computer science at Dartmouth College.
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Charles E. Leiserson is a computer scientist, specializing in the theory of parallel computing and distributed computing, and particularly practical applications thereof; as part of this effort, he developed the Cilk multithreaded language.
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Clifford Stein is a computer scientist, currently working as a professor at Columbia University in New York, NY. He earned his BSE from Princeton University in 1987, a MS from Massachusetts Institute of Technology in 1989, and a PhD from Massachusetts Institute of Technology in
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Introduction to Algorithms is a book by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. It is used as the textbook for algorithms courses at many universities.
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