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Ford-Fulkerson Algorithm: Maximizing Flow in Networks

Learn the Ford-Fulkerson algorithm to maximize flow in networks by finding augmenting paths, determining capacities, and updating residual capacities.

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Ford-Fulkerson Algorithm: Maximizing Flow in Networks

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  1. 15.082 and 6.855J The Ford-Fulkerson Augmenting Path Algorithm for the Maximum Flow Problem

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

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

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

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

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

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

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

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

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

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

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

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

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

  15. 1 2 5 1 1 2 2 s 4 t 2 2 3 Ford-Fulkerson Max Flow Here is the optimal flow

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