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The Ford-Fulkerson Augmenting Path Algorithm for
the Maximum Flow Problem
Thanks to Jim Orlin & MIT OCW
3
Ford-Fulkerson Max Flow
4
11
2
21
2
3
3
1
s
2
4
5
3
t
This is the original network, and the original residual network.
4
4
11
2
21
2
3
3
1
Ford-Fulkerson Max Flow
Find any s-t path in G(x)
s
2
4
5
3
t
5
4
1
1
2
13
Ford-Fulkerson Max Flow
Determine the capacity of the path.
Send units of flow in the path.Update residual capacities.
11
1
21
2
3
2
1
s
2
4
5
3
t
6
4
1
1
2
13
Ford-Fulkerson Max Flow
Find any s-t path
11
1
21
2
3
2
1
s
2
4
5
3
t
7
4
2
1
1
1
12
2
1
1
1
13
Ford-Fulkerson Max Flow
11
1
1
3
2
1
s
2
4
5
3
t
Determine the capacity of the path.
Send units of flow in the path.Update residual capacities.
8
4
2
1
1
1
12
2
1
1
1
13
Ford-Fulkerson Max Flow
11
1
1
3
2
1
s
2
4
5
3
t
Find any s-t path
9
11 1
11
4
1
2
1
12
11
3
Ford-Fulkerson Max Flow
113
2
1
s
2
4
5
3
t
Determine the capacity of the path.
Send units of flow in the path.Update residual capacities.
10
11 1
11
4
1
2
1
1
2
2
11
3
Ford-Fulkerson Max Flow
113
2
1
s
2
4
5
3
t
Find any s-t path
11
1
11
2 1 1
11
4
2
2
1
1
2
21
Ford-Fulkerson Max Flow
113
1
1
s
2
4
5
3
t
Determine the capacity of the path.
Send units of flow in the path.Update residual capacities.
2
12
11
2 1 1
11
4
2
2
1
1
2
21
Ford-Fulkerson Max Flow
113
1
1
s
2
4
5
3
t
Find any s-t path
2
13
111
11
4
13
11
2 1 1
3
2
21
21
Ford-Fulkerson Max Flow
2
1s
2
4
5
3
t
2
Determine the capacity of the path.
Send units of flow in the path.Update residual capacities.
14
111
11
4
13
11
2 1 1
3
2
21
21
Ford-Fulkerson Max Flow
2
1s
2
4
5
3
t
2
There is no s-t path in the residual network. This flow is optimal
15
111
11
4
13
11
2 1 1
3
2
21
21
Ford-Fulkerson Max Flow
2
1s
2
4
5
3
t
2
These are the nodes that are reachable from node s.
s
2
4
5
3
16
Ford-Fulkerson Max Flow
1
1
2
2
2
1
2s
2
4
5
3
t
Here is the optimal flow
