Algorithms, 4th Edition · Robert Sedgewick and Kevin Wayne · Copyright © 2002–2011 · February 23, 2012 4:05:54 PM

AlgorithmsF O U R T H E D I T I O N

R O B E R T S E D G E W I C K K E V I N W A Y N E

6.4 FORD-FULKERSON DEMO

click to begin demo

2

‣ Ford-Fulkerson algorithm‣ computing a min cut

Initialization. Start with 0 flow.

Ford-Fulkerson algorithm

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s t

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initialization

0 / 5 0

value of flow

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flow capacity

Augmenting path. Find an undirected path from s to t such that:

• Can increase flow on forward edges (not full).

• Can decrease flow on backward edge (not empty).

Idea: increase flow along augmenting paths

4

0 / 4

0 / 40 / 15

0 / 15

0 / 15 0 / 10

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0 / 8

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s t

0 / 10 0 / 15

0 / 100 / 5

1st augmenting path

0

0 / 10

Augmenting path. Find an undirected path from s to t such that:

• Can increase flow on forward edges (not full).

• Can decrease flow on backward edge (not empty).

Idea: increase flow along augmenting paths

4

0 / 4

0 / 40 / 15

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0 / 15 0 / 10

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s t

0 / 10 0 / 15

0 / 10

0 / 10 0 / 15

0 / 100 / 5

1st augmenting path

0

0 / 10

Augmenting path. Find an undirected path from s to t such that:

• Can increase flow on forward edges (not full).

• Can decrease flow on backward edge (not empty).

Idea: increase flow along augmenting paths

4

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0 / 15 0 / 10

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s t

0 / 10 0 / 15

0 / 10

0 / 10 0 / 15

0 / 100 / 5

1st augmenting path

0

0 / 10

bottleneck capacity = 10

Augmenting path. Find an undirected path from s to t such that:

• Can increase flow on forward edges (not full).

• Can decrease flow on backward edge (not empty).

Idea: increase flow along augmenting paths

4

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0 / 40 / 15

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0 / 15 0 / 10

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s t

0 / 10 0 / 15

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0 / 10 0 / 15

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10

10

10

—

—

—0 / 5

1st augmenting path

0 + 10 = 10

0 / 10

bottleneck capacity = 10

Augmenting path. Find an undirected path from s to t such that:

• Can increase flow on forward edges (not full).

• Can decrease flow on backward edge (not empty).

Idea: increase flow along augmenting paths

5

0 / 5

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0 / 4

0 / 15

0 / 15

0 / 10

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0 / 8

0 / 6

10 / 10 10 / 15

10 / 10s t

0 / 15 0 / 10

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2nd augmenting path

10

Augmenting path. Find an undirected path from s to t such that:

• Can increase flow on forward edges (not full).

• Can decrease flow on backward edge (not empty).

Idea: increase flow along augmenting paths

5

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10 / 10 10 / 15

10 / 10s t

0 / 15 0 / 10

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0 / 15 0 / 10

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2nd augmenting path

10

Augmenting path. Find an undirected path from s to t such that:

• Can increase flow on forward edges (not full).

• Can decrease flow on backward edge (not empty).

Idea: increase flow along augmenting paths

5

0 / 5

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0 / 4

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10 / 10 10 / 15

10 / 10s t

0 / 15 0 / 10

0 / 16

0 / 15 0 / 10

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10

10

10

—

—

—

2nd augmenting path

10 + 10 = 20

Augmenting path. Find an undirected path from s to t such that:

• Can increase flow on forward edges (not full).

• Can decrease flow on backward edge (not empty).

Idea: increase flow along augmenting paths

6

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0 / 4 0 / 150 / 6

10 / 10

10 / 10

10 / 15 10 / 10

10 / 16

s t0 / 5

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0 / 9

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10 / 15

3rd augmenting path

20

0 / 15

Augmenting path. Find an undirected path from s to t such that:

• Can increase flow on forward edges (not full).

• Can decrease flow on backward edge (not empty).

Idea: increase flow along augmenting paths

6

0 / 4

0 / 4 0 / 150 / 6

10 / 10

10 / 10

10 / 15 10 / 10

10 / 16

s t0 / 5

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10 / 15

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10 / 15

3rd augmenting path

20

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backward edge

(not empty)

Augmenting path. Find an undirected path from s to t such that:

• Can increase flow on forward edges (not full).

• Can decrease flow on backward edge (not empty).

Idea: increase flow along augmenting paths

6

0 / 4

0 / 4 0 / 150 / 6

10 / 10

10 / 10

10 / 15 10 / 10

10 / 16

s t0 / 5

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0 / 9

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10 / 15

0 / 5

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0 / 9

0 / 8

10 / 15

3rd augmenting path

20 + 5 = 255

—

5

—

5—

5—

5—

0 / 15

backward edge

(not empty)

Augmenting path. Find an undirected path from s to t such that:

• Can increase flow on forward edges (not full).

• Can decrease flow on backward edge (not empty).

Idea: increase flow along augmenting paths

7

0 / 4

0 / 4 0 / 15

10 / 10

10 / 105 / 5s t

0 / 6

5 / 10

5 / 9

5 / 8

5 / 15

10 / 1010 / 15

10 / 16

4th augmenting path

25

0 / 15

Augmenting path. Find an undirected path from s to t such that:

• Can increase flow on forward edges (not full).

• Can decrease flow on backward edge (not empty).

Idea: increase flow along augmenting paths

7

0 / 4

0 / 4 0 / 15

10 / 10

10 / 105 / 5s t

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5 / 10

5 / 9

5 / 8

0 / 6

5 / 10

5 / 9

5 / 8

5 / 155 / 15

10 / 1010 / 15

10 / 15

10 / 1610 / 16

4th augmenting path

25

0 / 15

backward edge

(not empty)

Augmenting path. Find an undirected path from s to t such that:

• Can increase flow on forward edges (not full).

• Can decrease flow on backward edge (not empty).

Idea: increase flow along augmenting paths

7

0 / 4

0 / 4 0 / 15

10 / 10

10 / 105 / 5s t

0 / 6

5 / 10

5 / 9

5 / 8

0 / 6

5 / 10

5 / 9

5 / 8

5 / 155 / 15

10 / 1010 / 15

10 / 15

10 / 1610 / 16

4th augmenting path

25 + 3 = 288

—

2—

8—

8—

13—

13

—

3—

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backward edge

(not empty)

Termination. All paths from s to t are blocked by either a

• Full forward edge.

• Empty backward edge.

Idea: increase flow along augmenting paths

8

no more augmenting paths

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10 / 105 / 5s t

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10 / 1013 / 15

28

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full forward edge

empty backward edge

9

‣ Ford-Fulkerson algorithm‣ computing a min cut

To compute mincut (A, B) from maxflow f :

• By augmenting path theorem, no augmenting paths with respect to f.

• Compute A = set of vertices connected to s by an undirected pathwith no full forward or empty backward edges.

Computing a mincut from a maxflow

10

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10 / 105 / 5 t

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s

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A

To compute mincut (A, B) from maxflow f :

• By augmenting path theorem, no augmenting paths with respect to f.

• Compute A = set of vertices connected to s by an undirected pathwith no full forward or empty backward edges.

Computing a mincut from a maxflow

10

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10 / 105 / 5 t

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ss

13 / 156 / 6

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A

To compute mincut (A, B) from maxflow f :

• By augmenting path theorem, no augmenting paths with respect to f.

• Compute A = set of vertices connected to s by an undirected pathwith no full forward or empty backward edges.

Computing a mincut from a maxflow

10

forward edge

(not full)

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10 / 105 / 5 t

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ss

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13 / 156 / 6

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A

To compute mincut (A, B) from maxflow f :

• By augmenting path theorem, no augmenting paths with respect to f.

• Compute A = set of vertices connected to s by an undirected pathwith no full forward or empty backward edges.

Computing a mincut from a maxflow

10

forward edge

(not full)

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10 / 105 / 5 t

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8 / 8

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10 / 10

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ss

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13 / 156 / 6

3 / 4

A

To compute mincut (A, B) from maxflow f :

• By augmenting path theorem, no augmenting paths with respect to f.

• Compute A = set of vertices connected to s by an undirected pathwith no full forward or empty backward edges.

Computing a mincut from a maxflow

10

forward edge

(not full)

backward edge

(not empty)

0 / 4

0 / 15

10 / 10

10 / 105 / 5 t

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8 / 8

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10 / 10

16 / 16

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ss

13 / 15

13 / 156 / 6

3 / 43 / 4

A

To compute mincut (A, B) from maxflow f :

• By augmenting path theorem, no augmenting paths with respect to f.

• Compute A = set of vertices connected to s by an undirected pathwith no full forward or empty backward edges.

Computing a mincut from a maxflow

10

forward edge

(not full)

backward edge

(not empty)

0 / 4

0 / 15

10 / 10

10 / 105 / 5 t

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8 / 9

8 / 8

2 / 15

10 / 10

16 / 16

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ss

13 / 15

13 / 156 / 6

3 / 43 / 4

A

To compute mincut (A, B) from maxflow f :

• By augmenting path theorem, no augmenting paths with respect to f.

• Compute A = set of vertices connected to s by an undirected pathwith no full forward or empty backward edges.

Computing a mincut from a maxflow

10

forward edge

(not full)

backward edge

(not empty)

0 / 4

0 / 15

10 / 10

10 / 105 / 5 t

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10 / 10

16 / 16

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ss

13 / 15

13 / 156 / 66 / 6

3 / 43 / 4

A

To compute mincut (A, B) from maxflow f :

• By augmenting path theorem, no augmenting paths with respect to f.

• Compute A = set of vertices connected to s by an undirected pathwith no full forward or empty backward edges.

Computing a mincut from a maxflow

10

forward edge

(not full)

backward edge

(not empty)

0 / 4

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10 / 10

10 / 105 / 5 t

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8 / 9

8 / 8

2 / 15

10 / 10

16 / 16

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ss

13 / 15

13 / 156 / 66 / 6

3 / 43 / 4

A

To compute mincut (A, B) from maxflow f :

• By augmenting path theorem, no augmenting paths with respect to f.

• Compute A = set of vertices connected to s by an undirected pathwith no full forward or empty backward edges.

Computing a mincut from a maxflow

10

forward edge

(not full)

backward edge

(not empty)

0 / 4

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10 / 10

10 / 105 / 5 t

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16 / 16

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ss

13 / 15

13 / 156 / 66 / 6

3 / 43 / 4

full forward edge

empty backward edge

A