Ford-Fulkerson algorithm - AbsoluteAstronomy.com

The Ford–Fulkerson Method (named for L. R. Ford, Jr.

L. R. Ford, Jr.

Lester Randolph Ford, Jr. is an American mathematician specializing in network flow problems. He is the son of mathematician Lester R. Ford, Sr..Ford's paper with D. R...

and D. R. Fulkerson

D. R. Fulkerson

Delbert Ray Fulkerson was a mathematician who co-developed the Ford-Fulkerson algorithm, one of the most well-known algorithms to solve the maximum flow problem in networks....

) computes the maximum flow

Maximum flow problem

In optimization theory, the maximum flow problem is to find a feasible flow through a single-source, single-sink flow network that is maximum....

in a flow network

Flow network

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 cannot exceed the capacity of the edge. Often in Operations Research, a directed graph is called a network, the vertices are called nodes and the edges are...

. It was published in 1956. The name "Ford–Fulkerson" is often also used for the Edmonds–Karp algorithm, which is a specialization of Ford–Fulkerson.

The idea behind the algorithm

Algorithm

In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

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

Let

be a graph, and for each edge from

, let

be the capacity and

be the flow. We want to find the maximum flow from the source

to the sink

. After every step in the algorithm the following is maintained:

Capacity constraints:		The flow along an edge can not exceed its capacity.
Skew symmetry:		The net flow from to must be the opposite of the net flow from to (see example).
Flow conservation:	$\sum_{w \in V} f(u,w) = 0$	That is, unless or . The net flow to a node is zero, except for the source, which "produces" flow, and the sink, which "consumes" flow.

This means that the flow through the network is a legal flow after each round in the algorithm. We define the residual network

to be the network with capacity

and no flow. Notice that it can happen that a flow from

is allowed in the residual
network, though disallowed in the original network: if

and

then

.

Algorithm Ford–Fulkerson

Inputs Graph

with flow capacity

, a source node

, and a sink node

Output A flow

from

which is a maximum

for all edges
While there is a path from to in , such that for all edges :
1. Find
2. For each edge
  1. (Send flow along the path)
  2. (The flow might be "returned" later)

The path in step 2 can be found with for example a breadth-first search

Breadth-first search

In graph theory, breadth-first search is a graph search algorithm that begins at the root node and explores all the neighboring nodes...

or a depth-first search

Depth-first search

Depth-first search is an algorithm for traversing or searching a tree, tree structure, or graph. One starts at the root and explores as far as possible along each branch before backtracking....

. If you use the former, the algorithm is called Edmonds–Karp.

When no more paths in step 2 can be found,

will not be able to reach

in the residual
network. If

is the set of nodes reachable by

in the residual network, then the total
capacity in the original network of edges from

to the remainder of

is on the one hand
equal to the total flow we found from

,
and on the other hand serves as an upper bound for all such flows.
This proves that the flow we found is maximal. See also Max-flow Min-cut theorem

Max-flow min-cut theorem

In optimization theory, the max-flow min-cut theorem states that in a flow network, the maximum amount of flow passing from the source to the sink is equal to the minimum capacity which when removed in a specific way from the network causes the situation that no flow can pass from the source to the...

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

(see big O notation

Big O notation

In mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann-Landau notation, or...

), where

is the number of edges in the graph and

is the maximum flow in the graph. This is because each augmenting path can be found in

time and increases the flow by an integer amount which is at least

.

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

time.

Integral example

The following example shows the first steps of Ford–Fulkerson in a flow network with 4 nodes, source

and sink

. This example shows the worst-case behaviour of the algorithm. In each step, only a flow of

is sent across the network. If breadth-first-search were used instead, only two steps would be needed.

Path	Capacity	Resulting flow network
Initial flow network


After 1998 more steps …
Final flow network

Notice how flow is "pushed back" from

when finding the path

Non-terminating example

Consider the flow network shown on the right, with source

, sink

, capacities of edges

and

respectively

and

and the capacity of all other edges some integer

. The constant

was chosen so, that

. We use augmenting paths according to the following table, where

and

Step	\| Augmenting path	\| Sent flow
Step	\| Augmenting path	\| Sent flow
0
1
2
3
4
5

Note that after step 1 as well as after step 5, the residual capacities of edges

and

are in the form

and

, respectively, for some

. This means that we can use augmenting paths

and

infinitely many times and residual capacities of these edges will always be in the same form. Total flow in the network after step 5 is

. If we continue to use augmenting paths as above, the total flow converges to

, while the maximum flow is

. In this case, the algorithm never terminates and the flow doesn't even converge to the maximum flow.

Python implementation

class Edge(object):
def __init__(self, u, v, w):
self.source = u
self.sink = v
self.capacity = w
def __repr__(self):
return "%s->%s:%s" % (self.source, self.sink, self.capacity)

class FlowNetwork(object):
def __init__(self):
self.adj = {}
self.flow = {}

def add_vertex(self, vertex):
self.adj[vertex] = []

def get_edges(self, v):
return self.adj[v]

def add_edge(self, u, v, w=0):
if u

v:
raise ValueError("u

v")
edge = Edge(u,v,w)
redge = Edge(v,u,0)
edge.redge = redge
redge.redge = edge
self.adj[u].append(edge)
self.adj[v].append(redge)
self.flow[edge] = 0
self.flow[redge] = 0

def find_path(self, source, sink, path):
if source sink:
return path
for edge in self.get_edges(source):
residual = edge.capacity - self.flow[edge]
if residual > 0 and not (edge,residual) in path:
result = self.find_path( edge.sink, sink, path + [(edge,residual)] )
if result != None:
return result

def max_flow(self, source, sink):
path = self.find_path(source, sink, [])
while path != None:
flow = min(res for edge,res in path)
for edge,res in path:
self.flow[edge] += flow
self.flow[edge.redge] -= flow
path = self.find_path(source, sink, [])
return sum(self.flow[edge] for edge in self.get_edges(source))

Usage example

For the example flow network in maximum flow problem

Maximum flow problem

In optimization theory, the maximum flow problem is to find a feasible flow through a single-source, single-sink flow network that is maximum....

we do the following:

g=FlowNetwork
map(g.add_vertex, ['s','o','p','q','r','t'])
g.add_edge('s','o',3)
g.add_edge('s','p',3)
g.add_edge('o','p',2)
g.add_edge('o','q',3)
g.add_edge('p','r',2)
g.add_edge('r','t',3)
g.add_edge('q','r',4)
g.add_edge('q','t',2)
print g.max_flow('s','t')

Output: 5
External links

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

Algorithm

Complexity

Integral example

Non-terminating example

Python implementation

v: raise ValueError("u

Usage example

v:
raise ValueError("u