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The Ford-Fulkerson Augmenting Path Algorithm for the Maximum Flow Problem Thanks to Jim Orlin & MIT OCW.

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Presentation on theme: "The Ford-Fulkerson Augmenting Path Algorithm for the Maximum Flow Problem Thanks to Jim Orlin & MIT OCW."— Presentation transcript:

1 The Ford-Fulkerson Augmenting Path Algorithm for the Maximum Flow Problem Thanks to Jim Orlin & MIT OCW

2 2 Ford-Fulkerson Max Flow 4 1 1 2 2 1 2 3 3 1 s 2 4 5 3 t This is the original network, plus reversals of the arcs.

3 3 Ford-Fulkerson Max Flow 4 1 1 2 2 1 2 3 3 1 s 2 4 5 3 t This is the original network, and the original residual network.

4 4 4 1 1 2 2 1 2 3 3 1 Ford-Fulkerson Max Flow Find any s-t path in G(x) s 2 4 5 3 t

5 5 4 1 1 2 1 3 Ford-Fulkerson Max Flow Determine the capacity  of the path. Send  units of flow in the path. Update residual capacities. 1 1 1 2 1 2 3 2 1 s 2 4 5 3 t

6 6 4 1 1 2 1 3 Ford-Fulkerson Max Flow Find any s-t path 1 1 1 2 1 2 3 2 1 s 2 4 5 3 t

7 7 4 2 1 1 1 1 2 2 1 1 1 1 3 Ford-Fulkerson Max Flow 1 1 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 8 4 2 1 1 1 1 2 2 1 1 1 1 3 Ford-Fulkerson Max Flow 1 1 1 1 3 2 1 s 2 4 5 3 t Find any s-t path

9 9 1 11 11 4 1 2 1 1 2 1 1 3 Ford-Fulkerson Max Flow 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.

10 10 1 11 11 4 1 2 1 1 2 2 1 1 3 Ford-Fulkerson Max Flow 1 1 3 2 1 s 2 4 5 3 t Find any s-t path

11 11 1 1 1 2 11 11 4 2 2 1 1 2 2 1 Ford-Fulkerson Max Flow 1 1 3 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 12 1 1 2 11 11 4 2 2 1 1 2 2 1 Ford-Fulkerson Max Flow 1 1 3 1 1 s 2 4 5 3 t Find any s-t path 2

13 13 1 1 1 1 1 4 13 1 1 2 11 3 2 2 1 2 1 Ford-Fulkerson Max Flow 2 1 s 2 4 5 3 t 2 Determine the capacity  of the path. Send  units of flow in the path. Update residual capacities.

14 14 1 1 1 1 1 4 13 1 1 2 11 3 2 2 1 2 1 Ford-Fulkerson Max Flow 2 1 s 2 4 5 3 t 2 There is no s-t path in the residual network. This flow is optimal

15 15 1 1 1 1 1 4 13 1 1 2 11 3 2 2 1 2 1 Ford-Fulkerson Max Flow 2 1 s 2 4 5 3 t 2 These are the nodes that are reachable from node s. s 2 4 5 3

16 16 Ford-Fulkerson Max Flow 1 1 2 2 2 1 2 s 2 4 5 3 t Here is the optimal flow

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