Transcript 15.082 & 6.855J & ESD.78J Algorithm Visualization The Ford-Fulkerson Augmenting Path Algorithm for the Maximum Flow Problem.

15.082 & 6.855J & ESD.78J
Algorithm Visualization
The Ford-Fulkerson Augmenting
Path Algorithm for the Maximum
Flow Problem
Ford-Fulkerson Max Flow
4
2
3
5
1
1
1
2
2
s
4
3
1
t
2
3
4
2
3
5
1 1
2
2
s
3
4
1
3
2
1
t
This is the original network,
and the original residual
network.
3
Ford-Fulkerson Max Flow
4
2
3
5
1
1
1
2
2
s
4
3
1
t
2
3
4
2
3
1 1
2
2
s
3
Find any s-t path in G(x)
5
4
1
3
2
1
t
4
Ford-Fulkerson Max Flow
4
2
3
5
1
1
2
1
2
s
4
2
3
1
1
2 1
t
1
3
Determine the capacity Δ of the path.
4
2
3
5
1 1
2
2
s
3
4
1
3
2
1
t
Send Δ units of flow in the path.
Update residual capacities.
5
Ford-Fulkerson Max Flow
4
2
3
5
1
1
2
1
1
2
s
4
2
3
1
2 1
t
1
3
4
2
3
5
1 1
2
2
s
3
4
1
3
2
Find any s-t path
1
t
6
Ford-Fulkerson Max Flow
4
2
5
3
1
1
2
1
s
2
3
11
4
1
1
1
3
3
5
1 1
2
2
s
3
4
1
3
2
2 1
1
t
1
Determine the capacity Δ of the path.
4
2
1
1
t
Send Δ units of flow in the path.
Update residual capacities.
7
Ford-Fulkerson Max Flow
4
2
5
3
1
1
2
1
1
11
s
2
3
4
1
1
1
3
2 1
1
t
1
4
2
3
5
1 1
2
2
s
3
4
1
3
2
Find any s-t path
1
t
8
Ford-Fulkerson Max Flow
4
2
5
3
1
1
1
s
2
3
11
4
1
2
1
1
3
3
5
1 1
2
2
s
3
4
1
3
2
1
1 2
t
1
Determine the capacity Δ of the path.
4
2
1
1
t
Send Δ units of flow in the path.
Update residual capacities.
9
Ford-Fulkerson Max Flow
4
2
5
3
1
1
1
2
1
11
s
2
3
4
1
2
1
1
3
1
1 2
t
1
4
2
3
5
1 1
2
2
s
3
4
1
3
2
Find any s-t path
1
t
10
Ford-Fulkerson Max Flow
4
2
5
3
1
1
1
1
2
s
1
2
11
4
1
2
1
2
1
3
3
5
1 1
2
2
s
3
4
1
3
2
1
2
2
1
Determine the capacity Δ of the path.
4
2
1
t
1
t
Send Δ units of flow in the path.
Update residual capacities.
11
Ford-Fulkerson Max Flow
4
2
5
3
1
1
1
2
1
11
s
1
2
4
1
2
1
2
t
1
2
1
3
2
1
4
2
3
5
1 1
2
2
s
3
4
1
3
2
Find any s-t path
1
t
12
Ford-Fulkerson Max Flow
4
3
2
1
3
2
1
5
1
1
1
s
1
2
1
4
1
2
1
2
1
3
3
5
1 1
2
2
s
3
4
1
3
2
2
1
Determine the capacity Δ of the path.
4
2
1
2
t
1
t
Send Δ units of flow in the path.
Update residual capacities.
13
Ford-Fulkerson Max Flow
4
3
2
1
3
2
1
5
1
1
1
s
1
2
1
4
1
2
1
2
1
3
4
2
3
5
1 1
2
2
s
3
4
1
3
2
1
t
1
2
t
2
1
There is no s-t path in
the residual network.
This flow is optimal
14
Ford-Fulkerson Max Flow
4
3
2
1
3
2
1
5
1
1
1
s
1
2
1
4
1
2
1
2
1
3
4
2
3
2
2
s
3
4
1
3
2
2
1
These are the nodes that
are reachable from node s.
5
1 1
1
2
t
1
t
15
Ford-Fulkerson Max Flow
1
2
5
1
1
2
2
s
4
2
t
2
3
4
2
3
1 1
2
2
s
3
Here is the
optimal flow
5
4
1
3
2
1
t
16
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15.082J / 6.855J / ESD.78J Network Optimization
Fall 2010
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