1 / 44

Lecture 13

Lecture 13. Introduction to Maximum Flows. Flow Network. The Ford Fulkerson Maximum Flow Algorithm. Begin x := 0; create the residual network G(x); while there is some directed path from s to t in G(x) do begin let P be a path from s to t in G(x); ∆ := δ (P);

jbullock
Download Presentation

Lecture 13

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lecture 13 Introduction to Maximum Flows

  2. Flow Network

  3. The Ford Fulkerson Maximum Flow Algorithm • Begin x := 0; • create the residual network G(x); • while there is some directed path from s to t in G(x) do • begin • let P be a path from s to t in G(x); • ∆:= δ(P); • send ∆ units of flow along P; • update the r's; • End • end {the flow x is now maximum}.

  4. The Ford-Fulkerson Augmenting Path Algorithm for the Maximum Flow Problem

  5. 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.

  6. 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)

  7. 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.

  8. 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

  9. 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.

  10. 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

  11. 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.

  12. 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

  13. 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.

  14. 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

  15. 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.

  16. 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

  17. 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.

  18. 1 2 5 1 1 2 2 s 4 t 2 2 3 Ford-Fulkerson Max Flow Here is the optimal flow

  19. Review for cutset theorem

  20. Quiz Sample

  21. Polynomial-time

More Related