Presentation is loading. Please wait.

Presentation is loading. Please wait.

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

Published byJulie Johnston Modified over 8 years ago

Similar presentations

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

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

Similar presentations

15.082 and 6.855J Cycle Canceling Algorithm. 2 A minimum cost flow problem 1 24 35 10, $4 20, $1 20, $2 25, $2 25, $5 20, $6 30, $7 25 0 0 0-25.

and 6.855J Cycle Canceling Algorithm. 2 A minimum cost flow problem , $4 20, $1 20, $2 25, $2 25, $5 20, $6 30, $
Maximum Flow and Minimum Cut Problems In this handout: Duality theory Upper bounds for maximum flow value Minimum Cut Problem Relationship between Maximum.

Maximum Flow and Minimum Cut Problems In this handout: Duality theory Upper bounds for maximum flow value Minimum Cut Problem Relationship between Maximum.
Outline LP formulation of minimal cost flow problem

Outline LP formulation of minimal cost flow problem
Max Flow Problem Given network N=(V,A), two nodes s,t of V, and capacities on the arcs: uij is the capacity on arc (i,j). Find non-negative flow fij for.

Max Flow Problem Given network N=(V,A), two nodes s,t of V, and capacities on the arcs: uij is the capacity on arc (i,j). Find non-negative flow fij for.
Chapter 6 Maximum Flow Problems Flows and Cuts Augmenting Path Algorithm.

Chapter 6 Maximum Flow Problems Flows and Cuts Augmenting Path Algorithm.
Network Optimization Models: Maximum Flow Problems

Network Optimization Models: Maximum Flow Problems
MAXIMUM FLOW Max-Flow Min-Cut Theorem (Ford Fukerson’s Algorithm)

MAXIMUM FLOW Max-Flow Min-Cut Theorem (Ford Fukerson’s Algorithm)
1 Augmenting Path Algorithm s 2 3 4 5t 10 9 8 4 6 2 0 0 0 0 0 0 0 0 G: Flow value = 0 0 flow capacity.

1 Augmenting Path Algorithm s t G: Flow value = 0 0 flow capacity.
1 Chapter 7 Network Flow Slides by Kevin Wayne. Copyright © 2005 Pearson-Addison Wesley. All rights reserved.

1 Chapter 7 Network Flow Slides by Kevin Wayne. Copyright © 2005 Pearson-Addison Wesley. All rights reserved.
1 Chapter 7 Network Flow Slides by Kevin Wayne. Copyright © 2005 Pearson-Addison Wesley. All rights reserved.

1 Chapter 7 Network Flow Slides by Kevin Wayne. Copyright © 2005 Pearson-Addison Wesley. All rights reserved.
Www.monash.edu.au 1 COMMONWEALTH OF AUSTRALIA Copyright Regulations 1969 WARNING This material has been reproduced and communicated to you by or on behalf.

1 COMMONWEALTH OF AUSTRALIA Copyright Regulations 1969 WARNING This material has been reproduced and communicated to you by or on behalf.
1 Augmenting Path Algorithm s 2 3 4 5t 10 9 8 4 6 2 0 0 0 0 0 0 0 0 G: Flow value = 0 0 flow capacity.

1 Augmenting Path Algorithm s t G: Flow value = 0 0 flow capacity.
Network Optimization Models: Maximum Flow Problems In this handout: The problem statement Solving by linear programming Augmenting path algorithm.

Network Optimization Models: Maximum Flow Problems In this handout: The problem statement Solving by linear programming Augmenting path algorithm.
3 t 4 1 2 s 2 1 4 23 2 1 4 3 4 1 the black numbers next to an arc is its capacity.

3 t s the black numbers next to an arc is its capacity.
Maximum Flows Lecture 4: Jan 19. Network transmission Given a directed graph G A source node s A sink node t Goal: To send as much information from s.

Maximum Flows Lecture 4: Jan 19. Network transmission Given a directed graph G A source node s A sink node t Goal: To send as much information from s.
The Dinitz Algorithm An example of a run. v1v1 Residual & Layered Networks construction s t v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 25 1114 9 5 10 8 1 5 8 9 7 6.

The Dinitz Algorithm An example of a run. v1v1 Residual & Layered Networks construction s t v2v2 v3v3 v4v4 v5v5 v6v6 v7v
CSE 421 Algorithms Richard Anderson Lecture 22 Network Flow.

CSE 421 Algorithms Richard Anderson Lecture 22 Network Flow.
NetworkModel-1 Network Optimization Models. NetworkModel-2 Network Terminology A network consists of a set of nodes and arcs. The arcs may have some flow.

NetworkModel-1 Network Optimization Models. NetworkModel-2 Network Terminology A network consists of a set of nodes and arcs. The arcs may have some flow.
15.082 and 6.855J The Capacity Scaling Algorithm.

and 6.855J The Capacity Scaling Algorithm.
Maximization of Network Survivability against Intelligent and Malicious Attacks (Cont’d) Presented by Erion Lin.

Maximization of Network Survivability against Intelligent and Malicious Attacks (Cont’d) Presented by Erion Lin.

Similar presentations

Ads by Google