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Introduction to Maximum Flows

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Presentation on theme: "Introduction to Maximum Flows"— Presentation transcript:

1 Introduction to Maximum Flows
Lecture 13 Introduction to Maximum Flows

2 Flow Network

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13 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}.

14 The Ford-Fulkerson Augmenting Path Algorithm for the Maximum Flow Problem
Obtain a network, and use the same network to illustrate the shortest path problem for communication networks, the max flow problem, the minimum cost flow problem, and the multicommodity flow problem. This will be a very efficient way of introducing the four problems. (Perhaps under 10 minutes of class time.)

15 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.

16 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)

17 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.

18 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

19 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.

20 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 Find any s-t path

21 Ford-Fulkerson Max Flow
4 2 5 1 3 1 1 1 1 1 s 4 t 1 2 2 1 3 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.

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

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

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

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

26 Ford-Fulkerson Max Flow
3 4 2 5 1 1 2 3 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

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

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

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33 Review for cutset theorem

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45 Quiz Sample

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52 Polynomial-time

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